Wikipedia Glossary of Category Theory.txt

"Introduction (Wikipedia Glossary of Category Theory)"¬"
<p>This deck has taken definitions outlined in <a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory"">Wikipedia's Glossary of category theory</a> and put them into a form which can be easily learnt/revised using <a href=""https://apps.ankiweb.net/"">Anki</a> a cross platform app specifically designed for long term knowledge retention.</p>
<h2>Notes</h2>
<p>Please note the modifications which have been made &amp; where you can find updates.</p>
<ol align=""left"">
<li>The front of every card has ""(Wikipedia Glossary of Category Theory)"" appended to the end so that if you have any other words in your collection, the Wikipedia definition will still be added when importing it.</li>
<li>Any updates or corrections to the deck will be available at <a href=""https://github.com/darigovresearch/Wikipedia-Glossary-of-Category-Theory-Flashcards"">https://github.com/darigovresearch/Wikipedia-Glossary-of-Category-Theory-Flashcards</a> so do return periodically to check if you have the latest version.</li>
</ol>
<p>Feel free to share the deck and give the repository a star so more people are likely to see this work and can get the most out of it.</p>
<h2>License</h2>
<p>Unless otherwise specified, everything in this deck is covered by the following licence:</p>
<p>This deck uses material from the Wikipedia article <a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory"">""Glossary of category theory""</a>, which is released under the <a href=""https://creativecommons.org/licenses/by-sa/3.0/"">Creative Commons Attribution-Share-Alike License 3.0</a>.</p>
<p>To see this work in full go to <a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory"">https://en.wikipedia.org/wiki/Glossary_of_category_theory</a></p>
"
"abelian (Wikipedia Glossary of Category Theory)"¬"
A category is <a href=""https://en.wikipedia.org/wiki/Abelian_category"" title=""Abelian category"">abelian</a> if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
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"accessible (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;Given a <a href=""https://en.wikipedia.org/wiki/Cardinal_number"" title=""Cardinal number"">cardinal number</a> κ, an object <i>X</i> in a category is <a href=""https://en.wikipedia.org/wiki/Accessible_object"" class=""mw-redirect"" title=""Accessible object"">κ-accessible</a> (or κ-compact or κ-presentable) if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} (X,-)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} (X,-)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ff04322e141ee468880e85236e7ed322d81f4b66"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:11.473ex; height:2.843ex;"" alt=""\operatorname {Hom} (X,-)""/></span> commutes with κ-filtered colimits.</dd>
<dd class=""glossary"">2.&#160;&#160;Given a <a href=""https://en.wikipedia.org/wiki/Regular_cardinal"" title=""Regular cardinal"">regular cardinal</a> κ, a category is <a href=""https://en.wikipedia.org/wiki/Accessible_category"" title=""Accessible category"">κ-accessible</a> if it has κ-filtered colimits and there exists a small set <i>S</i> of κ-compact objects that generates the category under colimits, meaning every object can be written as a colimit of diagrams of objects in <i>S</i>.</dd>
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"additive (Wikipedia Glossary of Category Theory)"¬"
A category is <a href=""https://en.wikipedia.org/wiki/Additive_category"" title=""Additive category"">additive</a> if it is preadditive (to be precise, has some pre-additive structure) and admits all finite <a href=""https://en.wikipedia.org/wiki/Coproduct"" title=""Coproduct"">coproducts</a>. Although ""preadditive"" is an additional structure, one can show ""additive"" is a <i>property</i> of a category; i.e., one can ask whether a given category is additive or not.<sup id=""cite_ref-2"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-2"">&#91;2&#93;</a></sup>
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"adjunction (Wikipedia Glossary of Category Theory)"¬"
An <a href=""https://en.wikipedia.org/wiki/Adjoint_functor"" class=""mw-redirect"" title=""Adjoint functor"">adjunction</a> (also called an adjoint pair) is a pair of functors <i>F</i>: <i>C</i> → <i>D</i>, <i>G</i>: <i>D</i> → <i>C</i> such that there is a ""natural"" bijection
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} _{D}(F(X),Y)\simeq \operatorname {Hom} _{C}(X,G(Y))}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>D</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mi>Y</mi>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2243;<!-- ≃ --></mo>
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>C</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>G</mi>
        <mo stretchy=""false"">(</mo>
        <mi>Y</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} _{D}(F(X),Y)\simeq \operatorname {Hom} _{C}(X,G(Y))}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/0258fb8922a809b57061db833bcc03e86c771255"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:36.235ex; height:2.843ex;"" alt=""\operatorname {Hom} _{D}(F(X),Y)\simeq \operatorname {Hom} _{C}(X,G(Y))""/></span>;</dd></dl>
<i>F</i> is said to be left adjoint to <i>G</i> and <i>G</i> to right adjoint to <i>F</i>. Here, ""natural"" means there is a natural isomorphism <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} _{D}(F(-),-)\simeq \operatorname {Hom} _{C}(-,G(-))}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>D</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2243;<!-- ≃ --></mo>
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>C</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo>,</mo>
        <mi>G</mi>
        <mo stretchy=""false"">(</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} _{D}(F(-),-)\simeq \operatorname {Hom} _{C}(-,G(-))}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/083a00b0f3c1b02fd1cc45d7ade02a9db2aaeea2"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:35.961ex; height:2.843ex;"" alt=""\operatorname {Hom} _{D}(F(-),-)\simeq \operatorname {Hom} _{C}(-,G(-))""/></span> of bifunctors (which are contravariant in the first variable.)
"
"algebra for a monad (Wikipedia Glossary of Category Theory)"¬"
Given a monad <i>T</i> in a category <i>X</i>, an <a href=""https://en.wikipedia.org/wiki/Algebra_for_a_monad"" class=""mw-redirect"" title=""Algebra for a monad"">algebra for <i>T</i></a> or a <i>T</i>-algebra is an object in <i>X</i> with a <a href=""https://en.wikipedia.org/wiki/Monoid_action"" class=""mw-redirect"" title=""Monoid action"">monoid action</a> of <i>T</i> (""algebra"" is misleading and ""<i>T</i>-object"" is perhaps a better term.) For example, given a group <i>G</i> that determines a monad <i>T</i> in <b>Set</b> in the standard way, a <i>T</i>-algebra is a set with an <a href=""https://en.wikipedia.org/wiki/Group_action_(mathematics)"" class=""mw-redirect"" title=""Group action (mathematics)"">action</a> of <i>G</i>.
"
"amnestic (Wikipedia Glossary of Category Theory)"¬"
A functor is amnestic if it has the property: if <i>k</i> is an isomorphism and <i>F</i>(<i>k</i>) is an identity, then <i>k</i> is an identity."
"balanced (Wikipedia Glossary of Category Theory)"¬"
A category is balanced if every bimorphism is an isomorphism.
"
"Beck's theorem (Wikipedia Glossary of Category Theory)"¬"
<a href=""https://en.wikipedia.org/wiki/Beck%27s_monadicity_theorem"" title=""Beck&#39;s monadicity theorem"">Beck's theorem</a> characterizes the category of <a href=""https://en.wikipedia.org/wiki/Algebra_for_a_monad"" class=""mw-redirect"" title=""Algebra for a monad"">algebras for a given monad</a>.
"
"bicategory (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Bicategory"" title=""Bicategory"">bicategory</a> is a model of a weak <a href=""https://en.wikipedia.org/wiki/2-category"" class=""mw-redirect"" title=""2-category"">2-category</a>.
"
"bifunctor (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Bifunctor"" class=""mw-redirect"" title=""Bifunctor"">bifunctor</a> from a pair of categories <i>C</i> and <i>D</i> to a category <i>E</i> is a functor <i>C</i> × <i>D</i> → <i>E</i>. For example, for any category <i>C</i>, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} (-,-)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo>,</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} (-,-)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/109100e375473e3b95c60a7ebf1adc71ce797c04"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:11.301ex; height:2.843ex;"" alt=""\operatorname {Hom} (-,-)""/></span> is a bifunctor from <i>C</i><sup>op</sup> and <i>C</i> to <b>Set</b>.
"
"bimorphism (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Bimorphism"" class=""mw-redirect"" title=""Bimorphism"">bimorphism</a> is a morphism that is both an epimorphism and a monomorphism.
"
"Bousfield localization (Wikipedia Glossary of Category Theory)"¬"
See <a href=""https://en.wikipedia.org/wiki/Bousfield_localization"" title=""Bousfield localization"">Bousfield localization</a>.
"
"calculus of functors (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Calculus_of_functors"" title=""Calculus of functors"">calculus of functors</a> is a technique of studying functors in the manner similar to the way a <a href=""https://en.wikipedia.org/wiki/Function_(mathematics)"" title=""Function (mathematics)"">function</a> is studied via its <a href=""https://en.wikipedia.org/wiki/Taylor_series"" title=""Taylor series"">Taylor series</a> expansion; whence, the term ""calculus"".
"
"cartesian closed (Wikipedia Glossary of Category Theory)"¬"
A category is <a href=""https://en.wikipedia.org/wiki/Cartesian_closed_category"" title=""Cartesian closed category"">cartesian closed</a> if it has a terminal object and that any two objects have a product and exponential.
"
"cartesian functor (Wikipedia Glossary of Category Theory)"¬"
Given relative categories <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle p:F\to C,q:G\to C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>p</mi>
        <mo>:</mo>
        <mi>F</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
        <mo>,</mo>
        <mi>q</mi>
        <mo>:</mo>
        <mi>G</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle p:F\to C,q:G\to C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/314cafcf4d6d17fbb197dbfb88bc89c86443d04c"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; margin-left: -0.089ex; width:21.565ex; height:2.509ex;"" alt=""{\displaystyle p:F\to C,q:G\to C}""/></span> over the same base category <i>C</i>, a functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:F\to G}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>F</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>G</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:F\to G}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/85edcca983d67148f3ef74994f52e81db9fa9a81"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:10.397ex; height:2.509ex;"" alt=""f:F\to G""/></span> over <i>C</i> is cartesian if it sends cartesian morphisms to cartesian morphisms.
"
"cartesian morphism (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;Given a functor π: <i>C</i> → <i>D</i> (e.g., a <a href=""https://en.wikipedia.org/wiki/Prestack"" title=""Prestack"">prestack</a> over schemes), a morphism <i>f</i>: <i>x</i> → <i>y</i> in <i>C</i> is <a href=""https://en.wikipedia.org/wiki/Cartesian_morphism"" class=""mw-redirect"" title=""Cartesian morphism"">π-cartesian</a> if, for each object <i>z</i> in <i>C</i>, each morphism <i>g</i>: <i>z</i> → <i>y</i> in <i>C</i> and each morphism <i>v</i>: π(<i>z</i>) → π(<i>x</i>) in <i>D</i> such that π(<i>g</i>) = π(<i>f</i>) ∘ <i>v</i>, there exists a unique morphism <i>u</i>: <i>z</i> → <i>x</i> such that π(<i>u</i>) = <i>v</i> and <i>g</i> = <i>f</i> ∘ <i>u</i>.</dd>
<dd class=""glossary"">2.&#160;&#160;Given a functor π: <i>C</i> → <i>D</i> (e.g., a <a href=""https://en.wikipedia.org/wiki/Prestack"" title=""Prestack"">prestack</a> over rings), a morphism <i>f</i>: <i>x</i> → <i>y</i> in <i>C</i> is <a href=""https://en.wikipedia.org/wiki/Cartesian_morphism"" class=""mw-redirect"" title=""Cartesian morphism"">π-coCartesian</a> if, for each object <i>z</i> in <i>C</i>, each morphism <i>g</i>: <i>x</i> → <i>z</i> in <i>C</i> and each morphism <i>v</i>: π(<i>y</i>) → π(<i>z</i>) in <i>D</i> such that π(<i>g</i>) = <i>v</i> ∘ π(<i>f</i>), there exists a unique morphism <i>u</i>: <i>y</i> → <i>z</i> such that π(<i>u</i>) = <i>v</i> and <i>g</i> = <i>u</i>  ∘ <i>f</i>. (In short, <i>f</i> is the dual of a π-cartesian morphism.)</dd>
"
"Cartesian square (Wikipedia Glossary of Category Theory)"¬"
A commutative diagram that is isomorphic to the diagram given as a fiber product.
"
"categorical logic (Wikipedia Glossary of Category Theory)"¬"
<a href=""https://en.wikipedia.org/wiki/Categorical_logic"" title=""Categorical logic"">Categorical logic</a> is an approach to <a href=""https://en.wikipedia.org/wiki/Mathematical_logic"" title=""Mathematical logic"">mathematical logic</a> that uses category theory.
"
"categorification (Wikipedia Glossary of Category Theory)"¬"
<a href=""https://en.wikipedia.org/wiki/Categorification"" title=""Categorification"">Categorification</a> is a process of replacing sets and set-theoretic concepts with categories and category-theoretic concepts in some nontrivial way to capture categoric flavors. Decategorification is the reverse of categorification.
"
"category (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Category_(mathematics)"" title=""Category (mathematics)"">category</a> consists of the following data
<ol><li>A class of objects,</li>
<li>For each pair of objects <i>X</i>, <i>Y</i>, a set <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} (X,Y)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>Y</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} (X,Y)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5ad87d5a22d6eadb0dafdabdd0f55237821860"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:11.438ex; height:2.843ex;"" alt=""\operatorname {Hom} (X,Y)""/></span>, whose elements are called morphisms from <i>X</i> to <i>Y</i>,</li>
<li>For each triple of objects <i>X</i>, <i>Y</i>, <i>Z</i>, a map (called composition)
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \circ :\operatorname {Hom} (Y,Z)\times \operatorname {Hom} (X,Y)\to \operatorname {Hom} (X,Z),\,(g,f)\mapsto g\circ f}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo>&#x2218;<!-- ∘ --></mo>
        <mo>:</mo>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>Y</mi>
        <mo>,</mo>
        <mi>Z</mi>
        <mo stretchy=""false"">)</mo>
        <mo>&#x00D7;<!-- × --></mo>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>Y</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>Z</mi>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mo stretchy=""false"">(</mo>
        <mi>g</mi>
        <mo>,</mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x21A6;<!-- ↦ --></mo>
        <mi>g</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \circ :\operatorname {Hom} (Y,Z)\times \operatorname {Hom} (X,Y)\to \operatorname {Hom} (X,Z),\,(g,f)\mapsto g\circ f}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/4779b7635d2828567e3d682af101abd927b3157a"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:58.338ex; height:2.843ex;"" alt=""\circ :\operatorname {Hom} (Y,Z)\times \operatorname {Hom} (X,Y)\to \operatorname {Hom} (X,Z),\,(g,f)\mapsto g\circ f""/></span>,</dd></dl></li>
<li>For each object <i>X</i>, an identity morphism <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {id} _{X}\in \operatorname {Hom} (X,X)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>id</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mo>&#x2208;<!-- ∈ --></mo>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {id} _{X}\in \operatorname {Hom} (X,X)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/02ac4475da908d9973b4d6bd4f5ea8920c131f6c"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:18.057ex; height:2.843ex;"" alt=""\operatorname {id} _{X}\in \operatorname {Hom} (X,X)""/></span></li></ol>
subject to the conditions: for any morphisms <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:X\to Y}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>X</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>Y</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:X\to Y}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:10.583ex; height:2.509ex;"" alt=""f:X\to Y""/></span>, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle g:Y\to Z}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>g</mi>
        <mo>:</mo>
        <mi>Y</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>Z</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle g:Y\to Z}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/44e7e6d2116baaa7aa88d1adbf796ade5997a237"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:10.121ex; height:2.509ex;"" alt=""g:Y\to Z""/></span> and <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle h:Z\to W}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>h</mi>
        <mo>:</mo>
        <mi>Z</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>W</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle h:Z\to W}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a637c09435fc1275a10137fe5295ad1b08c79a"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:11.006ex; height:2.176ex;"" alt=""h:Z\to W""/></span>,
<ul><li><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle (h\circ g)\circ f=h\circ (g\circ f)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo stretchy=""false"">(</mo>
        <mi>h</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>g</mi>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
        <mo>=</mo>
        <mi>h</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>g</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle (h\circ g)\circ f=h\circ (g\circ f)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ae58809efb94aa5c6610f70fdaca836662c87e30"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:22.963ex; height:2.843ex;"" alt=""(h\circ g)\circ f=h\circ (g\circ f)""/></span> and <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {id} _{Y}\circ f=f\circ \operatorname {id} _{X}=f}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>id</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>Y</mi>
          </mrow>
        </msub>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
        <mo>=</mo>
        <mi>f</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <msub>
          <mi>id</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mi>f</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {id} _{Y}\circ f=f\circ \operatorname {id} _{X}=f}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a408b6fb4ca80fd0734e88eadfd5deadbc1559"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:20.774ex; height:2.509ex;"" alt=""\operatorname {id} _{Y}\circ f=f\circ \operatorname {id} _{X}=f""/></span>.</li></ul>
For example, a <a href=""https://en.wikipedia.org/wiki/Partially_ordered_set"" title=""Partially ordered set"">partially ordered set</a> can be viewed as a category: the objects are the elements of the set and for each pair of objects <i>x</i>, <i>y</i>, there is a unique morphism <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle x\to y}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>x</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>y</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle x\to y}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/43570cdee5baeaebcf546338f77cf67c27a7344f"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:6.099ex; height:2.176ex;"" alt=""x\to y""/></span> if and only if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle x\leq y}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>x</mi>
        <mo>&#x2264;<!-- ≤ --></mo>
        <mi>y</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle x\leq y}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:5.584ex; height:2.343ex;"" alt=""x\leq y""/></span>; the associativity of composition means transitivity.
"
"category of categories (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Category_of_categories"" class=""mw-redirect"" title=""Category of categories"">category of (small) categories</a>, denoted by <b>Cat</b>, is a category where the objects are all the categories which are small with respect to some fixed universe and the morphisms are all the <a href=""https://en.wikipedia.org/wiki/Functor"" title=""Functor"">functors</a>.
"
"classifying space (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Classifying_space_of_a_category"" class=""mw-redirect"" title=""Classifying space of a category"">classifying space of a category</a> <i>C</i> is the geometric realization of the nerve of <i>C</i>.
"
"co (Wikipedia Glossary of Category Theory)"¬"
Often used synonymous with op-; for example, a <a href=""https://en.wikipedia.org/wiki/Colimit"" class=""mw-redirect"" title=""Colimit"">colimit</a> refers to an op-limit in the sense that it is a limit in the opposite category. But there might be a distinction; for example, an op-fibration is not the same thing as a <a href=""https://en.wikipedia.org/wiki/Cofibration"" title=""Cofibration"">cofibration</a>.
"
"coend (Wikipedia Glossary of Category Theory)"¬"
The coend of a functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F:C^{\text{op}}\times C\to X}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <mo>:</mo>
        <msup>
          <mi>C</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mtext>op</mtext>
          </mrow>
        </msup>
        <mo>&#x00D7;<!-- × --></mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F:C^{\text{op}}\times C\to X}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/7d29c0e3cc0f008cb97387d2fbe219cbf04cb459"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:17.645ex; height:2.343ex;"" alt=""F:C^{\text{op}}\times C\to X""/></span> is the dual of the <a href=""https://en.wikipedia.org/wiki/End_(category_theory)"" title=""End (category theory)"">end</a> of <i>F</i> and is denoted by
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \int ^{c\in C}F(c,c)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mo>&#x222B;<!-- ∫ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>c</mi>
            <mo>&#x2208;<!-- ∈ --></mo>
            <mi>C</mi>
          </mrow>
        </msup>
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo>,</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \int ^{c\in C}F(c,c)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/e79bba45ccf2e453f3bc647a0c40ffe8a984ab5d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -2.338ex; width:12.738ex; height:6.343ex;"" alt=""\int ^{c\in C}F(c,c)""/></span>.</dd></dl>
For example, if <i>R</i> is a ring, <i>M</i> a right <i>R</i>-module and <i>N</i> a left <i>R</i>-module, then the <a href=""https://en.wikipedia.org/wiki/Tensor_product_of_modules"" title=""Tensor product of modules"">tensor product</a> of <i>M</i> and <i>N</i> is
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle M\otimes _{R}N=\int ^{R}M\otimes _{\mathbb {Z} }N}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>M</mi>
        <msub>
          <mo>&#x2297;<!-- ⊗ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>R</mi>
          </mrow>
        </msub>
        <mi>N</mi>
        <mo>=</mo>
        <msup>
          <mo>&#x222B;<!-- ∫ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>R</mi>
          </mrow>
        </msup>
        <mi>M</mi>
        <msub>
          <mo>&#x2297;<!-- ⊗ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mrow class=""MJX-TeXAtom-ORD"">
              <mi mathvariant=""double-struck"">Z</mi>
            </mrow>
          </mrow>
        </msub>
        <mi>N</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle M\otimes _{R}N=\int ^{R}M\otimes _{\mathbb {Z} }N}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ecf1177fc9b9b96f6d69ead8e4505242c62b5bed"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -2.338ex; width:24.93ex; height:6.176ex;"" alt=""M\otimes _{R}N=\int ^{R}M\otimes _{\mathbb {Z} }N""/></span></dd></dl>
where <i>R</i> is viewed as a category with one object whose morphisms are the elements of <i>R</i>.
"
"coequalizer (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Coequalizer"" title=""Coequalizer"">coequalizer</a> of a pair of morphisms <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f,g:A\to B}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>,</mo>
        <mi>g</mi>
        <mo>:</mo>
        <mi>A</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>B</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f,g:A\to B}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/17911b4e8e6a2f58d32826d24e9f879918303975"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:12.487ex; height:2.509ex;"" alt=""f,g:A\to B""/></span> is the colimit of the pair. It is the dual of an equalizer.
"
"coherence theorem (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Coherence_theorem"" class=""mw-redirect"" title=""Coherence theorem"">coherence theorem</a> is a theorem of a form that states a weak structure is equivalent to a strict structure.
"
"coimage (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Coimage"" title=""Coimage"">coimage</a> of a morphism <i>f</i>: <i>X</i> → <i>Y</i> is the coequalizer of <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle X\times _{Y}X\rightrightarrows X}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>X</mi>
        <msub>
          <mo>&#x00D7;<!-- × --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>Y</mi>
          </mrow>
        </msub>
        <mi>X</mi>
        <mo stretchy=""false"">&#x21C9;<!-- ⇉ --></mo>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle X\times _{Y}X\rightrightarrows X}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/5e337c62274495d62f3a2675751d7ff1f9a2e415"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:13.881ex; height:2.509ex;"" alt=""{\displaystyle X\times _{Y}X\rightrightarrows X}""/></span>.
"
"colored operad (Wikipedia Glossary of Category Theory)"¬"
Another term for <a href=""https://en.wikipedia.org/wiki/Multicategory"" title=""Multicategory"">multicategory</a>, a generalized category where a morphism can have several domains. The notion of ""colored operad"" is more primitive than that of operad: in fact, an operad can be defined as a colored operad with a single object.
"
"comma (Wikipedia Glossary of Category Theory)"¬"
Given functors <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:C\to B,g:D\to B}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>B</mi>
        <mo>,</mo>
        <mi>g</mi>
        <mo>:</mo>
        <mi>D</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>B</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:C\to B,g:D\to B}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/e80026d3380949f95365a3cf6cd6f83b0743345d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:21.75ex; height:2.509ex;"" alt=""f:C\to B,g:D\to B""/></span>, the <a href=""https://en.wikipedia.org/wiki/Comma_category"" title=""Comma category"">comma category</a> <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle (f\downarrow g)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">&#x2193;<!-- ↓ --></mo>
        <mi>g</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle (f\downarrow g)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/691ef16e5c2043faa6240322119ce7c2b947c2cc"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:6.657ex; height:2.843ex;"" alt=""(f\downarrow g)""/></span> is a category where (1) the objects are morphisms <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f(c)\to g(d)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>g</mi>
        <mo stretchy=""false"">(</mo>
        <mi>d</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f(c)\to g(d)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f801affd490799e3b05130735e1235b7e4d06f00"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:11.85ex; height:2.843ex;"" alt=""f(c)\to g(d)""/></span> and (2) a morphism from <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \alpha :f(c)\to g(d)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>&#x03B1;<!-- α --></mi>
        <mo>:</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>g</mi>
        <mo stretchy=""false"">(</mo>
        <mi>d</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \alpha :f(c)\to g(d)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/4e28f0a4ccdbbdbe09710fa7986ee357f29a29f9"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:15.275ex; height:2.843ex;"" alt=""\alpha :f(c)\to g(d)""/></span> to <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \beta :f(c')\to g(d')}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>&#x03B2;<!-- β --></mi>
        <mo>:</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <msup>
          <mi>c</mi>
          <mo>&#x2032;</mo>
        </msup>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>g</mi>
        <mo stretchy=""false"">(</mo>
        <msup>
          <mi>d</mi>
          <mo>&#x2032;</mo>
        </msup>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \beta :f(c')\to g(d')}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f09aa1edb639e22568ce0c39958114f52da2600d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:16.491ex; height:3.009ex;"" alt=""\beta :f(c&#039;)\to g(d&#039;)""/></span> consists of <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle c\to c'}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>c</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msup>
          <mi>c</mi>
          <mo>&#x2032;</mo>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle c\to c'}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/0a72a1d1fa8c1a1a2a673a2da134afc53f9b4849"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:6.312ex; height:2.509ex;"" alt=""c\to c&#039;""/></span> and <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle d\to d'}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>d</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msup>
          <mi>d</mi>
          <mo>&#x2032;</mo>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle d\to d'}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/60b11d1aa3c8790aa1f05dd1628ad3dd098b3937"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:6.733ex; height:2.509ex;"" alt=""d\to d&#039;""/></span> such that <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f(c)\to f(c'){\overset {\beta }{\to }}g(d')}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <msup>
          <mi>c</mi>
          <mo>&#x2032;</mo>
        </msup>
        <mo stretchy=""false"">)</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mover>
            <mo stretchy=""false"">&#x2192;<!-- → --></mo>
            <mi>&#x03B2;<!-- β --></mi>
          </mover>
        </mrow>
        <mi>g</mi>
        <mo stretchy=""false"">(</mo>
        <msup>
          <mi>d</mi>
          <mo>&#x2032;</mo>
        </msup>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f(c)\to f(c'){\overset {\beta }{\to }}g(d')}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ca06473f47ea5fcd8cd9eca3081d01b11931c4"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:19.64ex; height:4.176ex;"" alt=""f(c)\to f(c&#039;){\overset {\beta }{\to }}g(d&#039;)""/></span> is <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f(c){\overset {\alpha }{\to }}g(d)\to g(d').}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mover>
            <mo stretchy=""false"">&#x2192;<!-- → --></mo>
            <mi>&#x03B1;<!-- α --></mi>
          </mover>
        </mrow>
        <mi>g</mi>
        <mo stretchy=""false"">(</mo>
        <mi>d</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>g</mi>
        <mo stretchy=""false"">(</mo>
        <msup>
          <mi>d</mi>
          <mo>&#x2032;</mo>
        </msup>
        <mo stretchy=""false"">)</mo>
        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f(c){\overset {\alpha }{\to }}g(d)\to g(d').}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/3dd5f607d9691e8d61f0fa044e46cc6088cb6d7b"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:19.649ex; height:3.676ex;"" alt=""f(c){\overset {\alpha }{\to }}g(d)\to g(d&#039;).""/></span> For example, if <i>f</i> is the identity functor and <i>g</i> is the constant functor with a value <i>b</i>, then it is the slice category of <i>B</i> over an object <i>b</i>.
"
"comonad (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Comonad"" class=""mw-redirect"" title=""Comonad"">comonad</a> in a category <i>X</i> is a <a href=""https://en.wikipedia.org/wiki/Comonoid"" class=""mw-redirect"" title=""Comonoid"">comonoid</a> in the monoidal category of endofunctors of <i>X</i>.
"
"compact (Wikipedia Glossary of Category Theory)"¬"
Probably synonymous with <a href=""#accessible"">#accessible</a>.
"
"complete (Wikipedia Glossary of Category Theory)"¬"
A category is <a href=""https://en.wikipedia.org/wiki/Complete_category"" title=""Complete category"">complete</a> if all small limits exist.
"
"composition (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;A composition of morphisms in a category is part of the datum defining the category.</dd>
<dd class=""glossary"">2.&#160;&#160;If <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:C\to D,\,g:D\to E}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>D</mi>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mi>g</mi>
        <mo>:</mo>
        <mi>D</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>E</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:C\to D,\,g:D\to E}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/5d7f22c41e911df6b8d689023cca869060e69fb0"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:22.309ex; height:2.509ex;"" alt=""{\displaystyle f:C\to D,\,g:D\to E}""/></span> are functors, then the composition <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle g\circ f}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>g</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle g\circ f}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:4.589ex; height:2.509ex;"" alt=""g\circ f""/></span> or <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle gf}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>g</mi>
        <mi>f</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle gf}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae6992a2a836c1ff200f058911a5a15f32de24c"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:2.395ex; height:2.509ex;"" alt=""gf""/></span> is the functor defined by: for an object <i>x</i> and a morphism <i>u</i> in <i>C</i>, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle (g\circ f)(x)=g(f(x)),\,(g\circ f)(u)=g(f(u))}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo stretchy=""false"">(</mo>
        <mi>g</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">(</mo>
        <mi>x</mi>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mi>g</mi>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>x</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mo stretchy=""false"">(</mo>
        <mi>g</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">(</mo>
        <mi>u</mi>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mi>g</mi>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>u</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle (g\circ f)(x)=g(f(x)),\,(g\circ f)(u)=g(f(u))}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/757795766ec492cf0785db1441437655d8382e3a"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:41.379ex; height:2.843ex;"" alt=""{\displaystyle (g\circ f)(x)=g(f(x)),\,(g\circ f)(u)=g(f(u))}""/></span>.</dd>
<dd class=""glossary"">3.&#160;&#160;Natural transformations are composed pointwise: if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varphi :f\to g,\,\psi :g\to h}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>&#x03C6;<!-- φ --></mi>
        <mo>:</mo>
        <mi>f</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>g</mi>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mi>&#x03C8;<!-- ψ --></mi>
        <mo>:</mo>
        <mi>g</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>h</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varphi :f\to g,\,\psi :g\to h}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/715466f80333923d902b98f97e33c5a67f9bcad3"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:20.406ex; height:2.676ex;"" alt=""{\displaystyle \varphi :f\to g,\,\psi :g\to h}""/></span> are natural transformations, then <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \psi \circ \varphi }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>&#x03C8;<!-- ψ --></mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>&#x03C6;<!-- φ --></mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \psi \circ \varphi }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/988077e1de2926c867d79a4cd08b73a74d0a8d69"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:5.228ex; height:2.676ex;"" alt=""\psi \circ \varphi ""/></span> is the natural transformation given by <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle (\psi \circ \varphi )_{x}=\psi _{x}\circ \varphi _{x}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo stretchy=""false"">(</mo>
        <mi>&#x03C8;<!-- ψ --></mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>&#x03C6;<!-- φ --></mi>
        <msub>
          <mo stretchy=""false"">)</mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>x</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <msub>
          <mi>&#x03C8;<!-- ψ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>x</mi>
          </mrow>
        </msub>
        <mo>&#x2218;<!-- ∘ --></mo>
        <msub>
          <mi>&#x03C6;<!-- φ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>x</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle (\psi \circ \varphi )_{x}=\psi _{x}\circ \varphi _{x}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f7cfb5a0b892385581dd46daf361c6cdc4dae1a6"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:18.881ex; height:2.843ex;"" alt=""{\displaystyle (\psi \circ \varphi )_{x}=\psi _{x}\circ \varphi _{x}}""/></span>.</dd>
"
"concrete (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Concrete_category"" title=""Concrete category"">concrete category</a> <i>C</i> is a category such that there is a faithful functor from <i>C</i> to <b><a href=""https://en.wikipedia.org/wiki/Category_of_sets"" title=""Category of sets"">Set</a></b>; e.g., <b><a href=""https://en.wikipedia.org/wiki/Category_of_vector_spaces"" class=""mw-redirect"" title=""Category of vector spaces"">Vec</a></b>, <b><a href=""https://en.wikipedia.org/wiki/Category_of_groups"" title=""Category of groups"">Grp</a></b> and <b><a href=""https://en.wikipedia.org/wiki/Category_of_topological_spaces"" title=""Category of topological spaces"">Top</a></b>.
"
"cone (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Cone_(category_theory)"" title=""Cone (category theory)"">cone</a> is a way to express the <a href=""https://en.wikipedia.org/wiki/Universal_property"" title=""Universal property"">universal property</a> of a colimit (or dually a limit). One can show<sup id=""cite_ref-3"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-3"">&#91;3&#93;</a></sup> that the colimit <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varinjlim }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-OP"">
          <munder>
            <mi>lim</mi>
            <mo>&#x2192;<!-- → --></mo>
          </munder>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varinjlim }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/0753732a44a7b3d11249c0b1f344a457a3cb7178"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.637ex; margin-bottom: -0.367ex; width:3.36ex; height:3.843ex;"" alt=""\varinjlim ""/></span> is the left adjoint to the diagonal functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \Delta :C\to \operatorname {Fct} (I,C)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
        <mo>:</mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>Fct</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>I</mi>
        <mo>,</mo>
        <mi>C</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \Delta :C\to \operatorname {Fct} (I,C)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/250205f85b5653e80b7ac70483c1064390f37aef"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:18.49ex; height:2.843ex;"" alt=""\Delta :C\to \operatorname {Fct} (I,C)""/></span>, which sends an object <i>X</i> to the constant functor with value <i>X</i>; that is, for any <i>X</i> and any functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:I\to C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>I</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:I\to C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/9271e0d23650020d206dcdb2277513962308597f"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:9.768ex; height:2.509ex;"" alt=""f:I\to C""/></span>,
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} (\varinjlim f,X)\simeq \operatorname {Hom} (f,\Delta _{X}),}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mrow class=""MJX-TeXAtom-OP"">
          <munder>
            <mi>lim</mi>
            <mo>&#x2192;<!-- → --></mo>
          </munder>
        </mrow>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>f</mi>
        <mo>,</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2243;<!-- ≃ --></mo>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo>,</mo>
        <msub>
          <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} (\varinjlim f,X)\simeq \operatorname {Hom} (f,\Delta _{X}),}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ef2f41a7c0f4f5927c41927d8efbe87343a81fcc"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.637ex; margin-bottom: -0.367ex; width:30.967ex; height:4.009ex;"" alt=""\operatorname {Hom} (\varinjlim f,X)\simeq \operatorname {Hom} (f,\Delta _{X}),""/></span></dd></dl>
provided the colimit in question exists. The right-hand side is then the set of cones with vertex <i>X</i>.<sup id=""cite_ref-4"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-4"">&#91;4&#93;</a></sup>
"
"connected (Wikipedia Glossary of Category Theory)"¬"
A category is <a href=""https://en.wikipedia.org/wiki/Connected_category"" title=""Connected category"">connected</a> if, for each pair of objects <i>x</i>, <i>y</i>, there exists a finite sequence of objects <i>z</i><sub><i>i</i></sub> such that <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle z_{0}=x,z_{n}=y}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>z</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>0</mn>
          </mrow>
        </msub>
        <mo>=</mo>
        <mi>x</mi>
        <mo>,</mo>
        <msub>
          <mi>z</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mi>y</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle z_{0}=x,z_{n}=y}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/fd70d0790a9a1a03e82973f3b21aa4d23ed38792"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:14.151ex; height:2.009ex;"" alt=""z_{0}=x,z_{n}=y""/></span> and either <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} (z_{i},z_{i+1})}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>z</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
        <mo>,</mo>
        <msub>
          <mi>z</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
        </msub>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} (z_{i},z_{i+1})}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/9c8f46bdbc28500556d70d0fcdd6cbde04af87d1"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:13.547ex; height:2.843ex;"" alt=""\operatorname {Hom} (z_{i},z_{i+1})""/></span> or <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} (z_{i+1},z_{i})}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>z</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
        </msub>
        <mo>,</mo>
        <msub>
          <mi>z</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} (z_{i+1},z_{i})}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/0851fa283f317f2944c2ad850ac6deaa0ce264a4"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:13.547ex; height:2.843ex;"" alt=""\operatorname {Hom} (z_{i+1},z_{i})""/></span> is nonempty for any <i>i</i>.
"
"conservative functor (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Conservative_functor"" title=""Conservative functor"">conservative functor</a> is a functor that reflects isomorphisms. Many forgetful functors are conservative, but the forgetful functor from <b>Top</b> to <b>Set</b> is not conservative.
"
"constant (Wikipedia Glossary of Category Theory)"¬"
A functor is <a href=""https://en.wikipedia.org/wiki/Constant_functor"" class=""mw-redirect"" title=""Constant functor"">constant</a> if it maps every object in a category to the same object <i>A</i> and every morphism to the identity on <i>A</i>. Put in another way, a functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:C\to D}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>D</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:C\to D}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f9af673473fce78b42b08129374eef6a040b35bb"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:10.52ex; height:2.509ex;"" alt=""f:C\to D""/></span> is constant if it factors as: <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle C\to \{A\}{\overset {i}{\to }}D}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mo fence=""false"" stretchy=""false"">{</mo>
        <mi>A</mi>
        <mo fence=""false"" stretchy=""false"">}</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mover>
            <mo stretchy=""false"">&#x2192;<!-- → --></mo>
            <mi>i</mi>
          </mover>
        </mrow>
        <mi>D</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle C\to \{A\}{\overset {i}{\to }}D}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ad83fc209e7a54578dc00f46e0a1c8709f6a0c"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:13.696ex; height:4.009ex;"" alt=""C\to \{A\}{\overset {i}{\to }}D""/></span> for some object <i>A</i> in <i>D</i>, where <i>i</i> is the inclusion of the discrete category { <i>A</i> }.
"
"contravariant functor (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Contravariant_functor"" class=""mw-redirect"" title=""Contravariant functor"">contravariant functor</a> <i>F</i> from a category <i>C</i> to a category <i>D</i> is a (covariant) functor from <i>C</i><sup>op</sup> to <i>D</i>. It is sometimes also called a <a href=""https://en.wikipedia.org/wiki/Presheaf_(category_theory)"" title=""Presheaf (category theory)"">presheaf</a> especially when <i>D</i> is <b>Set</b> or the variants. For example, for each set <i>S</i>, let <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle {\mathfrak {P}}(S)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-ORD"">
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""fraktur"">P</mi>
          </mrow>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>S</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle {\mathfrak {P}}(S)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/97e79fcd08ef084f1ce08775defb96803cd2a02c"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:5.233ex; height:2.843ex;"" alt=""{\mathfrak {P}}(S)""/></span> be the power set of <i>S</i> and for each function <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:S\to T}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>S</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>T</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:S\to T}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/0436c7b3237f256dfaa71c9a43025ff8e943d57d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:9.965ex; height:2.509ex;"" alt=""f:S\to T""/></span>, define
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle {\mathfrak {P}}(f):{\mathfrak {P}}(T)\to {\mathfrak {P}}(S)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-ORD"">
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""fraktur"">P</mi>
          </mrow>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
        <mo>:</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""fraktur"">P</mi>
          </mrow>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>T</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""fraktur"">P</mi>
          </mrow>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>S</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle {\mathfrak {P}}(f):{\mathfrak {P}}(T)\to {\mathfrak {P}}(S)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd2bd2b8b2f6629977dd61579f92f3175a61908"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:21.166ex; height:2.843ex;"" alt=""{\mathfrak {P}}(f):{\mathfrak {P}}(T)\to {\mathfrak {P}}(S)""/></span></dd></dl>
by sending a subset <i>A</i> of <i>T</i> to the pre-image <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f^{-1}(A)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>f</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>&#x2212;<!-- − --></mo>
            <mn>1</mn>
          </mrow>
        </msup>
        <mo stretchy=""false"">(</mo>
        <mi>A</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f^{-1}(A)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/7b636827df9e757c54e8fe4a2067fbcb6651d31f"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:7.206ex; height:3.176ex;"" alt=""f^{-1}(A)""/></span>. With this, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle {\mathfrak {P}}:\mathbf {Set} \to \mathbf {Set} }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-ORD"">
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""fraktur"">P</mi>
          </mrow>
        </mrow>
        <mo>:</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle {\mathfrak {P}}:\mathbf {Set} \to \mathbf {Set} }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/eb155a58659cf5b9bde9f6167994b6614797e2b0"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:14.975ex; height:2.676ex;"" alt=""{\mathfrak {P}}:\mathbf {Set} \to \mathbf {Set} ""/></span> is a contravariant functor.
"
"coproduct (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Coproduct"" title=""Coproduct"">coproduct</a> of a family of objects <i>X</i><sub><i>i</i></sub> in a category <i>C</i> indexed by a set <i>I</i> is the inductive limit <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varinjlim }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-OP"">
          <munder>
            <mi>lim</mi>
            <mo>&#x2192;<!-- → --></mo>
          </munder>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varinjlim }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/0753732a44a7b3d11249c0b1f344a457a3cb7178"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.637ex; margin-bottom: -0.367ex; width:3.36ex; height:3.843ex;"" alt=""\varinjlim ""/></span> of the functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle I\to C,\,i\mapsto X_{i}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>I</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mi>i</mi>
        <mo stretchy=""false"">&#x21A6;<!-- ↦ --></mo>
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle I\to C,\,i\mapsto X_{i}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/a1df6b6f6ee5f004136520fcd3742d2911f2a4fb"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:15.114ex; height:2.509ex;"" alt=""I\to C,\,i\mapsto X_{i}""/></span>, where <i>I</i> is viewed as a discrete category. It is the dual of the product of the family. For example, a coproduct in <b><a href=""https://en.wikipedia.org/wiki/Category_of_groups"" title=""Category of groups"">Grp</a></b> is a <a href=""https://en.wikipedia.org/wiki/Free_product"" title=""Free product"">free product</a>.
"
"core (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/w/index.php?title=Core_(category_theory)&amp;action=edit&amp;redlink=1"" class=""new"" title=""Core (category theory) (page does not exist)"">core</a> of a category is the maximal groupoid contained in the category.
"
"Day convolution (Wikipedia Glossary of Category Theory)"¬"
Given a group or monoid <i>M</i>, the <a href=""https://en.wikipedia.org/wiki/Day_convolution"" title=""Day convolution"">Day convolution</a> is the tensor product in <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \mathbf {Fct} (M,\mathbf {Set} )}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">F</mi>
          <mi mathvariant=""bold"">c</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>M</mi>
        <mo>,</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \mathbf {Fct} (M,\mathbf {Set} )}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd9ce311f71046a698eeca8803372cd8a4e51e2"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:12.945ex; height:2.843ex;"" alt=""\mathbf {Fct} (M,\mathbf {Set} )""/></span>.<sup id=""cite_ref-5"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-5"">&#91;5&#93;</a></sup>
"
"density theorem (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Density_theorem_(category_theory)"" title=""Density theorem (category theory)"">density theorem</a> states that every presheaf (a set-valued contravariant functor) is a colimit of representable presheaves. Yoneda's lemma embeds a category <i>C</i> into the category of presheaves on <i>C</i>. The density theorem then says the image is ""dense"", so to say. The name ""density"" is because of the analogy with the <a href=""https://en.wikipedia.org/wiki/Jacobson_density_theorem"" title=""Jacobson density theorem"">Jacobson density theorem</a> (or other variants) in abstract algebra.
"
"diagonal functor (Wikipedia Glossary of Category Theory)"¬"
Given categories <i>I</i>, <i>C</i>, the <a href=""https://en.wikipedia.org/wiki/Diagonal_functor"" title=""Diagonal functor"">diagonal functor</a> is the functor
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \Delta :C\to \mathbf {Fct} (I,C),\,A\mapsto \Delta _{A}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
        <mo>:</mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">F</mi>
          <mi mathvariant=""bold"">c</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>I</mi>
        <mo>,</mo>
        <mi>C</mi>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mi>A</mi>
        <mo stretchy=""false"">&#x21A6;<!-- ↦ --></mo>
        <msub>
          <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>A</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \Delta :C\to \mathbf {Fct} (I,C),\,A\mapsto \Delta _{A}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/441aeea7b0ff1018b409a87be3ccf9aa4581db6e"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:29.124ex; height:2.843ex;"" alt=""\Delta :C\to \mathbf {Fct} (I,C),\,A\mapsto \Delta _{A}""/></span></dd></dl>
that sends each object <i>A</i> to the constant functor with value <i>A</i> and each morphism <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:A\to B}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>A</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>B</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:A\to B}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/20040a52d9391f2fe271f0aaa300bf7887a0c7b1"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:10.337ex; height:2.509ex;"" alt=""f:A\to B""/></span> to the natural transformation <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \Delta _{f,i}:\Delta _{A}(i)=A\to \Delta _{B}(i)=B}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>f</mi>
            <mo>,</mo>
            <mi>i</mi>
          </mrow>
        </msub>
        <mo>:</mo>
        <msub>
          <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>A</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mi>A</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msub>
          <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>B</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mi>B</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \Delta _{f,i}:\Delta _{A}(i)=A\to \Delta _{B}(i)=B}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/09dae270f29276c39c04981f58825b2d3320b874"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.005ex; width:31.392ex; height:3.009ex;"" alt=""\Delta _{f,i}:\Delta _{A}(i)=A\to \Delta _{B}(i)=B""/></span> that is <i>f</i> at each <i>i</i>.
"
"diagram (Wikipedia Glossary of Category Theory)"¬"
Given a category <i>C</i>, a <a href=""https://en.wikipedia.org/wiki/Diagram_(category_theory)"" title=""Diagram (category theory)"">diagram</a> in <i>C</i> is a functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:I\to C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>I</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:I\to C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/9271e0d23650020d206dcdb2277513962308597f"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:9.768ex; height:2.509ex;"" alt=""f:I\to C""/></span> from a small category <i>I</i>.
"
"differential graded category (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Differential_graded_category"" title=""Differential graded category"">differential graded category</a> is a category whose Hom sets are equipped with structures of <a href=""https://en.wikipedia.org/wiki/Differential_graded_module"" title=""Differential graded module"">differential graded modules</a>. In particular, if the category has only one object, it is the same as a differential graded module.
"
"direct limit (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Direct_limit"" title=""Direct limit"">direct limit</a> is the <a href=""https://en.wikipedia.org/wiki/Colimit"" class=""mw-redirect"" title=""Colimit"">colimit</a> of a <a href=""https://en.wikipedia.org/wiki/Direct_system"" class=""mw-redirect"" title=""Direct system"">direct system</a>.
"
"discrete (Wikipedia Glossary of Category Theory)"¬"
A category is <a href=""https://en.wikipedia.org/wiki/Discrete_category"" title=""Discrete category"">discrete</a> if each morphism is an identity morphism (of some object). For example, a set can be viewed as a discrete category.
"
"distributor (Wikipedia Glossary of Category Theory)"¬"
Another term for ""profunctor"".
"
"Dwyer–Kan equivalence (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/w/index.php?title=Dwyer%E2%80%93Kan_equivalence&amp;action=edit&amp;redlink=1"" class=""new"" title=""Dwyer–Kan equivalence (page does not exist)"">Dwyer–Kan equivalence</a> is a generalization of an equivalence of categories to the simplicial context.<sup id=""cite_ref-6"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-6"">&#91;6&#93;</a></sup>"
"Eilenberg–Moore category (Wikipedia Glossary of Category Theory)"¬"
Another name for the category of <a href=""https://en.wikipedia.org/wiki/Algebra_for_a_monad"" class=""mw-redirect"" title=""Algebra for a monad"">algebras for a given monad</a>.
"
"empty (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Empty_category_(category_theory)"" class=""mw-redirect"" title=""Empty category (category theory)"">empty category</a> is a category with no object. It is the same thing as the <a href=""https://en.wikipedia.org/wiki/Empty_set"" title=""Empty set"">empty set</a> when the empty set is viewed as a discrete category.
"
"end (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/End_(category_theory)"" title=""End (category theory)"">end</a> of a functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F:C^{\text{op}}\times C\to X}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <mo>:</mo>
        <msup>
          <mi>C</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mtext>op</mtext>
          </mrow>
        </msup>
        <mo>&#x00D7;<!-- × --></mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F:C^{\text{op}}\times C\to X}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/7d29c0e3cc0f008cb97387d2fbe219cbf04cb459"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:17.645ex; height:2.343ex;"" alt=""F:C^{\text{op}}\times C\to X""/></span> is the limit
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \int _{c\in C}F(c,c)=\varprojlim (F^{\#}:C^{\#}\to X)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mo>&#x222B;<!-- ∫ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>c</mi>
            <mo>&#x2208;<!-- ∈ --></mo>
            <mi>C</mi>
          </mrow>
        </msub>
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo>,</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mrow class=""MJX-TeXAtom-OP"">
          <munder>
            <mi>lim</mi>
            <mo>&#x2190;<!-- ← --></mo>
          </munder>
        </mrow>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <msup>
          <mi>F</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0023;<!-- # --></mi>
          </mrow>
        </msup>
        <mo>:</mo>
        <msup>
          <mi>C</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0023;<!-- # --></mi>
          </mrow>
        </msup>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \int _{c\in C}F(c,c)=\varprojlim (F^{\#}:C^{\#}\to X)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/4a3637f54f1f8d3807a167d31e3f66eb38ef1399"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -2.338ex; width:34.245ex; height:5.676ex;"" alt=""\int _{c\in C}F(c,c)=\varprojlim (F^{\#}:C^{\#}\to X)""/></span></dd></dl>
where <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle C^{\#}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>C</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0023;<!-- # --></mi>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle C^{\#}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/76e791bc8a8ec24d48fe04d103cd6ceac7d91264"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:3.399ex; height:2.676ex;"" alt=""C^{\#}""/></span> is the category (called the <a href=""https://en.wikipedia.org/w/index.php?title=Subdivision_category&amp;action=edit&amp;redlink=1"" class=""new"" title=""Subdivision category (page does not exist)"">subdivision category</a> of <i>C</i>) whose objects are symbols <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle c^{\#},u^{\#}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>c</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0023;<!-- # --></mi>
          </mrow>
        </msup>
        <mo>,</mo>
        <msup>
          <mi>u</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0023;<!-- # --></mi>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle c^{\#},u^{\#}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/4525e27234bb37216663e58a4c494db5faf89fa7"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:6.573ex; height:3.009ex;"" alt=""c^{\#},u^{\#}""/></span> for all objects <i>c</i> and all morphisms <i>u</i> in <i>C</i> and whose morphisms are <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle b^{\#}\to u^{\#}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>b</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0023;<!-- # --></mi>
          </mrow>
        </msup>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msup>
          <mi>u</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0023;<!-- # --></mi>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle b^{\#}\to u^{\#}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/36a9aa346ce70967486a35c29909ef841df89389"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:9.144ex; height:2.676ex;"" alt=""b^{\#}\to u^{\#}""/></span> and <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle u^{\#}\to c^{\#}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>u</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0023;<!-- # --></mi>
          </mrow>
        </msup>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msup>
          <mi>c</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0023;<!-- # --></mi>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle u^{\#}\to c^{\#}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/445f8fefcfa36d990bcc0722d67425570c2c3479"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:9.153ex; height:2.676ex;"" alt=""u^{\#}\to c^{\#}""/></span> if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle u:b\to c}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>u</mi>
        <mo>:</mo>
        <mi>b</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>c</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle u:b\to c}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e11d100339876b14689ab04374021e4683d192"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:8.885ex; height:2.176ex;"" alt=""u:b\to c""/></span> and where <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F^{\#}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>F</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0023;<!-- # --></mi>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F^{\#}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/70f93d55ac14e1a22e4c1a6b4d728a4bac21a5ab"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:3.416ex; height:2.676ex;"" alt=""F^{\#}""/></span> is induced by <i>F</i> so that <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle c^{\#}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>c</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0023;<!-- # --></mi>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle c^{\#}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9cb1ae2001a45d3b59180cf605de0c3f788213"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:2.608ex; height:2.676ex;"" alt=""c^{\#}""/></span> would go to <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F(c,c)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo>,</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F(c,c)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/20b0138be348098e26a796e870a69f9fc355e040"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:6.598ex; height:2.843ex;"" alt=""F(c,c)""/></span> and <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle u^{\#},u:b\to c}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>u</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0023;<!-- # --></mi>
          </mrow>
        </msup>
        <mo>,</mo>
        <mi>u</mi>
        <mo>:</mo>
        <mi>b</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>c</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle u^{\#},u:b\to c}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/b0fecbf2b575c50a476a2ec165a460e53663af09"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:12.85ex; height:3.009ex;"" alt=""u^{\#},u:b\to c""/></span> would go to <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F(b,c)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>b</mi>
        <mo>,</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F(b,c)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/6644010c68869a377698c908017ba167d602376d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:6.588ex; height:2.843ex;"" alt=""F(b,c)""/></span>. For example, for functors <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F,G:C\to X}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <mo>,</mo>
        <mi>G</mi>
        <mo>:</mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F,G:C\to X}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/bd16f8e936ee4620154a13ed87c48b2b6c6cf165"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:13.899ex; height:2.509ex;"" alt=""F,G:C\to X""/></span>,
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \int _{c\in C}\operatorname {Hom} (F(c),G(c))}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mo>&#x222B;<!-- ∫ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>c</mi>
            <mo>&#x2208;<!-- ∈ --></mo>
            <mi>C</mi>
          </mrow>
        </msub>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mi>G</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \int _{c\in C}\operatorname {Hom} (F(c),G(c))}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/85cf0a5a6aec09700ea29be3f20b8f6d16247734"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -2.338ex; width:21.854ex; height:5.676ex;"" alt=""\int _{c\in C}\operatorname {Hom} (F(c),G(c))""/></span></dd></dl>
is the set of natural transformations from <i>F</i> to <i>G</i>. For more examples, see <a rel=""nofollow"" class=""external text"" href=""https://mathoverflow.net/q/78471"">this mathoverflow thread</a>. The dual of an end is a coend.
"
"endofunctor (Wikipedia Glossary of Category Theory)"¬"
A functor between the same category.
"
"enriched category (Wikipedia Glossary of Category Theory)"¬"
Given a monoidal category (<i>C</i>, ⊗, 1), a <a href=""https://en.wikipedia.org/wiki/Category_enriched"" class=""mw-redirect"" title=""Category enriched"">category enriched</a> over <i>C</i> is, informally, a category whose Hom sets are in <i>C</i>. More precisely, a category <i>D</i> enriched over <i>C</i> is a data consisting of
<ol><li>A class of objects,</li>
<li>For each pair of objects <i>X</i>, <i>Y</i> in <i>D</i>, an object <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Map} _{D}(X,Y)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>Map</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>D</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>Y</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Map} _{D}(X,Y)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ed9fe2e2ad1016bcb589421adadf9555d9b8b2ac"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:12.775ex; height:2.843ex;"" alt=""{\displaystyle \operatorname {Map} _{D}(X,Y)}""/></span> in <i>C</i>, called the <a href=""https://en.wikipedia.org/w/index.php?title=Mapping_object&amp;action=edit&amp;redlink=1"" class=""new"" title=""Mapping object (page does not exist)"">mapping object</a> from <i>X</i> to <i>Y</i>,</li>
<li>For each triple of objects <i>X</i>, <i>Y</i>, <i>Z</i> in <i>D</i>, a morphism in <i>C</i>,
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \circ :\operatorname {Map} _{D}(Y,Z)\otimes \operatorname {Map} _{D}(X,Y)\to \operatorname {Map} _{D}(X,Z)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo>&#x2218;<!-- ∘ --></mo>
        <mo>:</mo>
        <msub>
          <mi>Map</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>D</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>Y</mi>
        <mo>,</mo>
        <mi>Z</mi>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2297;<!-- ⊗ --></mo>
        <msub>
          <mi>Map</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>D</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>Y</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msub>
          <mi>Map</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>D</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>Z</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \circ :\operatorname {Map} _{D}(Y,Z)\otimes \operatorname {Map} _{D}(X,Y)\to \operatorname {Map} _{D}(X,Z)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ee45f19fef49a1a3d5c90915ef985b3b8926d739"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:47.488ex; height:2.843ex;"" alt=""{\displaystyle \circ :\operatorname {Map} _{D}(Y,Z)\otimes \operatorname {Map} _{D}(X,Y)\to \operatorname {Map} _{D}(X,Z)}""/></span>,</dd>
<dd>called the composition,</dd></dl></li>
<li>For each object <i>X</i> in <i>D</i>, a morphism <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle 1_{X}:1\to \operatorname {Map} _{D}(X,X)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mn>1</mn>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mo>:</mo>
        <mn>1</mn>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msub>
          <mi>Map</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>D</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle 1_{X}:1\to \operatorname {Map} _{D}(X,X)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa9cc409d56636d58610189805d7db82ba4e723"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:22.491ex; height:2.843ex;"" alt=""{\displaystyle 1_{X}:1\to \operatorname {Map} _{D}(X,X)}""/></span> in <i>C</i>, called the unit morphism of <i>X</i></li></ol>
subject to the conditions that (roughly) the compositions are associative and the unit morphisms act as the multiplicative identity.

For example, a category enriched over sets is an ordinary category.
"
"epimorphism (Wikipedia Glossary of Category Theory)"¬"
A morphism <i>f</i> is an <a href=""https://en.wikipedia.org/wiki/Epimorphism"" title=""Epimorphism"">epimorphism</a> if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle g=h}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>g</mi>
        <mo>=</mo>
        <mi>h</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle g=h}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/50f4fd70b1200ffb96f9b7dccc5b64296e8da9f4"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:5.553ex; height:2.509ex;"" alt=""g=h""/></span> whenever <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle g\circ f=h\circ f}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>g</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
        <mo>=</mo>
        <mi>h</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle g\circ f=h\circ f}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/6684a08ba3c9d443fa7fd91c5da2831846cd4f4e"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:12.5ex; height:2.509ex;"" alt=""g\circ f=h\circ f""/></span>. In other words, <i>f</i> is the dual of a monomorphism.
"
"equalizer (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Equalizer_(mathematics)"" class=""mw-redirect"" title=""Equalizer (mathematics)"">equalizer</a> of a pair of morphisms <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f,g:A\to B}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>,</mo>
        <mi>g</mi>
        <mo>:</mo>
        <mi>A</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>B</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f,g:A\to B}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/17911b4e8e6a2f58d32826d24e9f879918303975"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:12.487ex; height:2.509ex;"" alt=""f,g:A\to B""/></span> is the limit of the pair. It is the dual of a coequalizer.
"
"equivalence (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;A functor is an <a href=""https://en.wikipedia.org/wiki/Equivalence_of_categories"" title=""Equivalence of categories"">equivalence</a> if it is faithful, full and essentially surjective.</dd>
<dd class=""glossary"">2.&#160;&#160;A morphism in an ∞-category <i>C</i>  is an equivalence if it gives an isomorphism in the homotopy category of <i>C</i>.</dd>
"
"equivalent (Wikipedia Glossary of Category Theory)"¬"
A category is equivalent to another category if there is an <a href=""https://en.wikipedia.org/wiki/Equivalence_of_categories"" title=""Equivalence of categories"">equivalence</a> between them.
"
"essentially surjective (Wikipedia Glossary of Category Theory)"¬"
A functor <i>F</i> is called <a href=""https://en.wikipedia.org/w/index.php?title=Essentially_surjective&amp;action=edit&amp;redlink=1"" class=""new"" title=""Essentially surjective (page does not exist)"">essentially surjective</a> (or isomorphism-dense) if for every object <i>B</i> there exists an object <i>A</i> such that <i>F</i>(<i>A</i>) is isomorphic to <i>B</i>.
"
"evaluation (Wikipedia Glossary of Category Theory)"¬"
Given categories <i>C</i>, <i>D</i> and an object <i>A</i> in <i>C</i>, the <a href=""https://en.wikipedia.org/w/index.php?title=Evaluation_(category_theory)&amp;action=edit&amp;redlink=1"" class=""new"" title=""Evaluation (category theory) (page does not exist)"">evaluation</a> at <i>A</i> is the functor
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \mathbf {Fct} (C,D)\to D,\,\,F\mapsto F(A).}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">F</mi>
          <mi mathvariant=""bold"">c</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>C</mi>
        <mo>,</mo>
        <mi>D</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>D</mi>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mspace width=""thinmathspace"" />
        <mi>F</mi>
        <mo stretchy=""false"">&#x21A6;<!-- ↦ --></mo>
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>A</mi>
        <mo stretchy=""false"">)</mo>
        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \mathbf {Fct} (C,D)\to D,\,\,F\mapsto F(A).}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/baf3f750d84d4ff1cef0ce11e3f13902d2dda7c0"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:29.085ex; height:2.843ex;"" alt=""{\displaystyle \mathbf {Fct} (C,D)\to D,\,\,F\mapsto F(A).}""/></span></dd></dl>
For example, the <a href=""https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms"" title=""Eilenberg–Steenrod axioms"">Eilenberg–Steenrod axioms</a> give an instance when the functor is an equivalence."
"faithful (Wikipedia Glossary of Category Theory)"¬"
A functor is <a href=""https://en.wikipedia.org/wiki/Faithful_functor"" class=""mw-redirect"" title=""Faithful functor"">faithful</a> if it is injective when restricted to each <a href=""https://en.wikipedia.org/wiki/Hom-set"" class=""mw-redirect"" title=""Hom-set"">hom-set</a>.
"
"fundamental category (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/w/index.php?title=Fundamental_category_functor&amp;action=edit&amp;redlink=1"" class=""new"" title=""Fundamental category functor (page does not exist)"">fundamental category functor</a> <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \tau _{1}:s\mathbf {Set} \to \mathbf {Cat} }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>&#x03C4;<!-- τ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo>:</mo>
        <mi>s</mi>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">C</mi>
          <mi mathvariant=""bold"">a</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \tau _{1}:s\mathbf {Set} \to \mathbf {Cat} }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/8b55f7a1ab2510d355c14fa19d09f6d1111677cf"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:16.732ex; height:2.509ex;"" alt=""{\displaystyle \tau _{1}:s\mathbf {Set} \to \mathbf {Cat} }""/></span> is the left adjoint to the nerve functor <i>N</i>. For every category <i>C</i>, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \tau _{1}NC=C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>&#x03C4;<!-- τ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>1</mn>
          </mrow>
        </msub>
        <mi>N</mi>
        <mi>C</mi>
        <mo>=</mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \tau _{1}NC=C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ff414cc668473834cdaa7cb9b0c021e8eea3252c"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:10.765ex; height:2.509ex;"" alt=""{\displaystyle \tau _{1}NC=C}""/></span>.
"
"fundamental groupoid (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Fundamental_groupoid"" title=""Fundamental groupoid"">fundamental groupoid</a> <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \Pi _{1}X}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi mathvariant=""normal"">&#x03A0;<!-- Π --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>1</mn>
          </mrow>
        </msub>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \Pi _{1}X}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f019b75bc660071136e303138c75032784a2d11b"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:4.777ex; height:2.509ex;"" alt=""{\displaystyle \Pi _{1}X}""/></span> of a Kan complex <i>X</i> is the category where an object is a 0-simplex (vertex) <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \Delta ^{0}\to X}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>0</mn>
          </mrow>
        </msup>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \Delta ^{0}\to X}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/c069c90286c9e2e8eba1c80a2f916df76fe4bce0"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:8.584ex; height:2.676ex;"" alt=""{\displaystyle \Delta ^{0}\to X}""/></span>, a morphism is a homotopy class of a 1-simplex (path) <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \Delta ^{1}\to X}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>1</mn>
          </mrow>
        </msup>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \Delta ^{1}\to X}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/d5f957f28ce7df560e9b8f38489abc0e663e7239"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:8.584ex; height:2.676ex;"" alt=""{\displaystyle \Delta ^{1}\to X}""/></span> and a composition is determined by the Kan property.
"
"fibered category (Wikipedia Glossary of Category Theory)"¬"
A functor π: <i>C</i> → <i>D</i> is said to exhibit <i>C</i> as a <a href=""https://en.wikipedia.org/wiki/Fibered_category"" class=""mw-redirect"" title=""Fibered category"">category fibered over</a> <i>D</i> if, for each morphism <i>g</i>: <i>x</i> → π(<i>y</i>) in <i>D</i>, there exists a π-cartesian morphism <i>f</i>: <i>x'</i> → <i>y</i> in <i>C</i> such that π(<i>f</i>) = <i>g</i>. If <i>D</i> is the category of affine schemes (say of finite type over some field), then π is more commonly called a <a href=""https://en.wikipedia.org/wiki/Prestack"" title=""Prestack"">prestack</a>. <b>Note</b>: π is often a forgetful functor and in fact the <a href=""https://en.wikipedia.org/wiki/Grothendieck_construction"" title=""Grothendieck construction"">Grothendieck construction</a> implies that every fibered category can be taken to be that form (up to equivalences in a suitable sense).
"
"fiber product (Wikipedia Glossary of Category Theory)"¬"
Given a category <i>C</i> and a set <i>I</i>, the <a href=""https://en.wikipedia.org/wiki/Fiber_product"" class=""mw-redirect"" title=""Fiber product"">fiber product</a> over an object <i>S</i> of a family of objects <i>X</i><sub><i>i</i></sub> in <i>C</i> indexed by <i>I</i> is the product of the family in the <a href=""https://en.wikipedia.org/wiki/Slice_category"" class=""mw-redirect"" title=""Slice category"">slice category</a> <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle C_{/S}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>C</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mrow class=""MJX-TeXAtom-ORD"">
              <mo>/</mo>
            </mrow>
            <mi>S</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle C_{/S}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3892fe3773b4e868fa3e03c7063e7698e9c457"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.171ex; width:3.776ex; height:3.009ex;"" alt=""C_{/S}""/></span> of <i>C</i> over <i>S</i> (provided there are <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle X_{i}\to S}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>S</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle X_{i}\to S}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/49432f3b0e36081523010eb74e3c3eb064d0d919"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:7.837ex; height:2.509ex;"" alt=""X_{i}\to S""/></span>). The fiber product of two objects <i>X</i> and <i>Y</i> over an object <i>S</i> is denoted by <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle X\times _{S}Y}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>X</mi>
        <msub>
          <mo>&#x00D7;<!-- × --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>S</mi>
          </mrow>
        </msub>
        <mi>Y</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle X\times _{S}Y}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/420355afb0bca3f6051b50875bb0cc841b549eb8"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:7.886ex; height:2.509ex;"" alt=""X\times _{S}Y""/></span> and is also called a <a href=""https://en.wikipedia.org/wiki/Cartesian_square"" class=""mw-redirect"" title=""Cartesian square"">Cartesian square</a>.
"
"filtered (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;A <a href=""https://en.wikipedia.org/wiki/Filtered_category"" title=""Filtered category"">filtered category</a> (also called a filtrant category) is a nonempty category with the properties (1) given objects <i>i</i> and <i>j</i>, there are an object <i>k</i> and morphisms <i>i</i> → <i>k</i> and <i>j</i> → <i>k</i> and (2) given morphisms <i>u</i>, <i>v</i>: <i>i</i> → <i>j</i>, there are an object <i>k</i> and a morphism <i>w</i>: <i>j</i> → <i>k</i> such that <i>w</i> ∘ <i>u</i> = <i>w</i> ∘ <i>v</i>. A category <i>I</i> is filtered if and only if, for each finite category <i>J</i> and functor <i>f</i>: <i>J</i> → <i>I</i>, the set <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varprojlim \operatorname {Hom} (f(j),i)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-OP"">
          <munder>
            <mi>lim</mi>
            <mo>&#x2190;<!-- ← --></mo>
          </munder>
        </mrow>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>j</mi>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varprojlim \operatorname {Hom} (f(j),i)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f88ebf341bdcdf3bb993a6d22a51acb022b2b3b2"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.637ex; margin-bottom: -0.367ex; width:16.345ex; height:4.009ex;"" alt=""\varprojlim \operatorname {Hom} (f(j),i)""/></span> is nonempty for some object <i>i</i> in <i>I</i>.</dd>
<dd class=""glossary"">2.&#160;&#160;Given a cardinal number π, a category is said to be π-filtrant if, for each category <i>J</i> whose set of morphisms has cardinal number strictly less than π, the set <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varprojlim \operatorname {Hom} (f(j),i)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-OP"">
          <munder>
            <mi>lim</mi>
            <mo>&#x2190;<!-- ← --></mo>
          </munder>
        </mrow>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>j</mi>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varprojlim \operatorname {Hom} (f(j),i)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f88ebf341bdcdf3bb993a6d22a51acb022b2b3b2"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.637ex; margin-bottom: -0.367ex; width:16.345ex; height:4.009ex;"" alt=""\varprojlim \operatorname {Hom} (f(j),i)""/></span> is nonempty for some object <i>i</i> in <i>I</i>.</dd>
"
"finitary monad (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/w/index.php?title=Finitary_monad&amp;action=edit&amp;redlink=1"" class=""new"" title=""Finitary monad (page does not exist)"">finitary monad</a> or an algebraic monad is a monad on <b>Set</b> whose underlying endofunctor commutes with filtered colimits.
"
"finite (Wikipedia Glossary of Category Theory)"¬"
A category is finite if it has only finitely many morphisms.
"
"forgetful functor (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Forgetful_functor"" title=""Forgetful functor"">forgetful functor</a> is, roughly, a functor that loses some of data of the objects; for example, the functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \mathbf {Grp} \to \mathbf {Set} }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">G</mi>
          <mi mathvariant=""bold"">r</mi>
          <mi mathvariant=""bold"">p</mi>
        </mrow>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \mathbf {Grp} \to \mathbf {Set} }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/30dbb1a1256ddef855212e8e5d28764c4d22e0ef"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:12.052ex; height:2.509ex;"" alt=""\mathbf {Grp} \to \mathbf {Set} ""/></span> that sends a group to its underlying set and a group homomorphism to itself is a forgetful functor.
"
"free functor (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Free_functor"" class=""mw-redirect"" title=""Free functor"">free functor</a> is a left adjoint to a forgetful functor. For example, for a ring <i>R</i>, the functor that sends a set <i>X</i> to the <a href=""https://en.wikipedia.org/wiki/Free_module"" title=""Free module"">free <i>R</i>-module</a> generated by <i>X</i> is a free functor (whence the name).
"
"Frobenius category (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Frobenius_category"" title=""Frobenius category"">Frobenius category</a> is an <a href=""https://en.wikipedia.org/wiki/Exact_category"" title=""Exact category"">exact category</a> that has enough injectives and enough projectives and such that the class of injective objects coincides with that of projective objects.
"
"Fukaya category (Wikipedia Glossary of Category Theory)"¬"
See <a href=""https://en.wikipedia.org/wiki/Fukaya_category"" title=""Fukaya category"">Fukaya category</a>.
"
"full (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;A functor is <a href=""https://en.wikipedia.org/wiki/Full_functor"" class=""mw-redirect"" title=""Full functor"">full</a> if it is surjective when restricted to each <a href=""https://en.wikipedia.org/wiki/Hom-set"" class=""mw-redirect"" title=""Hom-set"">hom-set</a>.</dd>
<dd class=""glossary"">2.&#160;&#160;A category <i>A</i> is a <a href=""https://en.wikipedia.org/wiki/Full_subcategory"" class=""mw-redirect"" title=""Full subcategory"">full subcategory</a> of a category <i>B</i> if the inclusion functor from <i>A</i> to <i>B</i> is full.</dd>
"
"functor (Wikipedia Glossary of Category Theory)"¬"
Given categories <i>C</i>, <i>D</i>, a <a href=""https://en.wikipedia.org/wiki/Functor"" title=""Functor"">functor</a> <i>F</i> from <i>C</i> to <i>D</i> is a structure-preserving map from <i>C</i> to <i>D</i>; i.e., it consists of an object <i>F</i>(<i>x</i>) in <i>D</i> for each object <i>x</i> in <i>C</i> and a morphism <i>F</i>(<i>f</i>) in <i>D</i> for each morphism <i>f</i> in <i>C</i> satisfying the conditions: (1) <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F(f\circ g)=F(f)\circ F(g)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>g</mi>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>g</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F(f\circ g)=F(f)\circ F(g)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/9c5c4ae596dc0b30c6a779cc39471db28cac903e"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:22.927ex; height:2.843ex;"" alt=""{\displaystyle F(f\circ g)=F(f)\circ F(g)}""/></span> whenever <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f\circ g}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>g</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f\circ g}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:4.589ex; height:2.509ex;"" alt=""f\circ g""/></span> is defined and (2) <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F(\operatorname {id} _{x})=\operatorname {id} _{F(x)}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>id</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>x</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <msub>
          <mi>id</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>F</mi>
            <mo stretchy=""false"">(</mo>
            <mi>x</mi>
            <mo stretchy=""false"">)</mo>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F(\operatorname {id} _{x})=\operatorname {id} _{F(x)}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d17204c1492d281cc3ce7865a98cd56bea6846"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.171ex; width:15.383ex; height:3.176ex;"" alt=""F(\operatorname {id} _{x})=\operatorname {id} _{F(x)}""/></span>. For example,
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle {\mathfrak {P}}:\mathbf {Set} \to \mathbf {Set} ,\,S\mapsto {\mathfrak {P}}(S)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-ORD"">
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""fraktur"">P</mi>
          </mrow>
        </mrow>
        <mo>:</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mi>S</mi>
        <mo stretchy=""false"">&#x21A6;<!-- ↦ --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""fraktur"">P</mi>
          </mrow>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>S</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle {\mathfrak {P}}:\mathbf {Set} \to \mathbf {Set} ,\,S\mapsto {\mathfrak {P}}(S)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/7e5e4231623cd57f33418736ef65f2df574e6329"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:26.742ex; height:2.843ex;"" alt=""{\mathfrak {P}}:\mathbf {Set} \to \mathbf {Set} ,\,S\mapsto {\mathfrak {P}}(S)""/></span>,</dd></dl>
where <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle {\mathfrak {P}}(S)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-ORD"">
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""fraktur"">P</mi>
          </mrow>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>S</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle {\mathfrak {P}}(S)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/97e79fcd08ef084f1ce08775defb96803cd2a02c"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:5.233ex; height:2.843ex;"" alt=""{\mathfrak {P}}(S)""/></span> is the <a href=""https://en.wikipedia.org/wiki/Power_set"" title=""Power set"">power set</a> of <i>S</i> is a functor if we define: for each function <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:S\to T}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>S</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>T</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:S\to T}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/0436c7b3237f256dfaa71c9a43025ff8e943d57d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:9.965ex; height:2.509ex;"" alt=""f:S\to T""/></span>, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle {\mathfrak {P}}(f):{\mathfrak {P}}(S)\to {\mathfrak {P}}(T)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-ORD"">
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""fraktur"">P</mi>
          </mrow>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
        <mo>:</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""fraktur"">P</mi>
          </mrow>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>S</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""fraktur"">P</mi>
          </mrow>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>T</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle {\mathfrak {P}}(f):{\mathfrak {P}}(S)\to {\mathfrak {P}}(T)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/3ca3f48392b5dcbfe118af0d0849686313269cd9"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:21.166ex; height:2.843ex;"" alt=""{\mathfrak {P}}(f):{\mathfrak {P}}(S)\to {\mathfrak {P}}(T)""/></span> by <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle {\mathfrak {P}}(f)(A)=f(A)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-ORD"">
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""fraktur"">P</mi>
          </mrow>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">(</mo>
        <mi>A</mi>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>A</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle {\mathfrak {P}}(f)(A)=f(A)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/4aa8d94921420c9c30ff08098234c57745020e5c"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:16.494ex; height:2.843ex;"" alt=""{\mathfrak {P}}(f)(A)=f(A)""/></span>.
"
"functor category (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Functor_category"" title=""Functor category"">functor category</a> <b>Fct</b>(<i>C</i>, <i>D</i>) or <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle D^{C}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>D</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>C</mi>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle D^{C}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/108456f7fa636edcc6227f68cd48cade449ab356"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:3.406ex; height:2.676ex;"" alt=""{\displaystyle D^{C}}""/></span> from a category <i>C</i> to a category <i>D</i> is the category where the objects are all the functors from <i>C</i> to <i>D</i> and the morphisms are all the natural transformations between the functors.
"
"Gabriel–Popescu theorem (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Gabriel%E2%80%93Popescu_theorem"" title=""Gabriel–Popescu theorem"">Gabriel–Popescu theorem</a> says an abelian category is a <a href=""https://en.wikipedia.org/wiki/Serre_quotient_category"" class=""mw-redirect"" title=""Serre quotient category"">quotient</a> of the category of modules.
"
"generator (Wikipedia Glossary of Category Theory)"¬"
In a category <i>C</i>, a family of objects <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle G_{i},i\in I}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>G</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
        <mo>,</mo>
        <mi>i</mi>
        <mo>&#x2208;<!-- ∈ --></mo>
        <mi>I</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle G_{i},i\in I}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/872d74f282ba24214d0828ec8e3682a636b0b6c7"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:8.475ex; height:2.509ex;"" alt=""G_{i},i\in I""/></span> is a <a href=""https://en.wikipedia.org/wiki/Generator_(category_theory)"" title=""Generator (category theory)"">system of generators</a> of <i>C</i> if the functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle X\mapsto \prod _{i\in I}\operatorname {Hom} (G_{i},X)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>X</mi>
        <mo stretchy=""false"">&#x21A6;<!-- ↦ --></mo>
        <munder>
          <mo>&#x220F;<!-- ∏ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
            <mo>&#x2208;<!-- ∈ --></mo>
            <mi>I</mi>
          </mrow>
        </munder>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>G</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
        <mo>,</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle X\mapsto \prod _{i\in I}\operatorname {Hom} (G_{i},X)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/c7dac787fdc4e4d81c8ebae1e714fa84057c09ca"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -3.171ex; width:21.242ex; height:5.676ex;"" alt=""X\mapsto \prod _{i\in I}\operatorname {Hom} (G_{i},X)""/></span> is conservative. Its dual is called a system of cogenerators.
"
"Grothendieck's Galois theory (Wikipedia Glossary of Category Theory)"¬"
A category-theoretic generalization of <a href=""https://en.wikipedia.org/wiki/Galois_theory"" title=""Galois theory"">Galois theory</a>; see <a href=""https://en.wikipedia.org/wiki/Grothendieck%27s_Galois_theory"" title=""Grothendieck&#39;s Galois theory"">Grothendieck's Galois theory</a>.
"
"Grothendieck category (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Grothendieck_category"" title=""Grothendieck category"">Grothendieck category</a> is a certain well-behaved kind of an abelian category.
"
"Grothendieck construction (Wikipedia Glossary of Category Theory)"¬"
Given a functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle U:C\to \mathbf {Cat} }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>U</mi>
        <mo>:</mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">C</mi>
          <mi mathvariant=""bold"">a</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle U:C\to \mathbf {Cat} }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/75b03cd393ad3644f3c186164e5ce294e34147f3"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:13.37ex; height:2.176ex;"" alt=""U:C\to \mathbf {Cat} ""/></span>, let <i>D</i><sub><i>U</i></sub> be the category where the objects are pairs (<i>x</i>, <i>u</i>) consisting of an object <i>x</i> in <i>C</i> and an object <i>u</i> in the category <i>U</i>(<i>x</i>) and a morphism from (<i>x</i>, <i>u</i>) to (<i>y</i>, <i>v</i>) is a pair consisting of a morphism <i>f</i>: <i>x</i> → <i>y</i> in <i>C</i> and a morphism <i>U</i>(<i>f</i>)(<i>u</i>) → <i>v</i> in <i>U</i>(<i>y</i>). The passage from <i>U</i> to <i>D</i><sub><i>U</i></sub> is then called the <a href=""https://en.wikipedia.org/wiki/Grothendieck_construction"" title=""Grothendieck construction"">Grothendieck construction</a>.
"
"Grothendieck fibration (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Fibered_category"" class=""mw-redirect"" title=""Fibered category"">fibered category</a>.
"
"groupoid (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;A category is called a <a href=""https://en.wikipedia.org/wiki/Groupoid"" title=""Groupoid"">groupoid</a> if every morphism in it is an isomorphism.</dd>
<dd class=""glossary"">2.&#160;&#160;An ∞-category is called an <a href=""https://en.wikipedia.org/wiki/%E2%88%9E-groupoid"" title=""∞-groupoid"">∞-groupoid</a> if every morphism in it is an equivalence (or equivalently if it is a <a href=""https://en.wikipedia.org/wiki/Kan_complex"" class=""mw-redirect"" title=""Kan complex"">Kan complex</a>.)</dd>
"
"Hall algebra of a category (Wikipedia Glossary of Category Theory)"¬"
See <a href=""https://en.wikipedia.org/wiki/Ringel%E2%80%93Hall_algebra"" title=""Ringel–Hall algebra"">Ringel–Hall algebra</a>.
"
"heart (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Heart_(category_theory)"" class=""mw-redirect"" title=""Heart (category theory)"">heart</a> of a <a href=""https://en.wikipedia.org/wiki/T-structure"" title=""T-structure"">t-structure</a> (<span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle D^{\geq 0}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>D</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>&#x2265;<!-- ≥ --></mo>
            <mn>0</mn>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle D^{\geq 0}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/4b1b7169f0879440bcc3d98ca194de0e8a045409"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:4.257ex; height:2.676ex;"" alt=""{\displaystyle D^{\geq 0}}""/></span>, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle D^{\leq 0}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>D</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>&#x2264;<!-- ≤ --></mo>
            <mn>0</mn>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle D^{\leq 0}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/09393cfd188cb7ada9e88b4671ab4ceddd253427"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:4.257ex; height:2.676ex;"" alt=""D^{{\leq 0}}""/></span>) on a triangulated category is the intersection <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle D^{\geq 0}\cap D^{\leq 0}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>D</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>&#x2265;<!-- ≥ --></mo>
            <mn>0</mn>
          </mrow>
        </msup>
        <mo>&#x2229;<!-- ∩ --></mo>
        <msup>
          <mi>D</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>&#x2264;<!-- ≤ --></mo>
            <mn>0</mn>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle D^{\geq 0}\cap D^{\leq 0}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8d9e73492542263789ad64537c74ab18496d4f"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:11.097ex; height:2.676ex;"" alt=""{\displaystyle D^{\geq 0}\cap D^{\leq 0}}""/></span>. It is an abelian category.
"
"Higher category theory (Wikipedia Glossary of Category Theory)"¬"
<a href=""https://en.wikipedia.org/wiki/Higher_category_theory"" title=""Higher category theory"">Higher category theory</a> is a subfield of category theory that concerns the study of <a href=""https://en.wikipedia.org/wiki/N-category"" class=""mw-redirect"" title=""N-category""><i>n</i>-categories</a> and <a href=""https://en.wikipedia.org/wiki/%E2%88%9E-categories"" class=""mw-redirect"" title=""∞-categories"">∞-categories</a>.
"
"homological dimension (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Homological_dimension"" title=""Homological dimension"">homological dimension</a> of an abelian category with enough injectives is the least non-negativer integer <i>n</i> such that every object in the category admits an injective resolution of length at most <i>n</i>. The dimension is ∞ if no such integer exists. For example, the homological dimension of <a href=""https://en.wikipedia.org/wiki/Category_of_modules"" title=""Category of modules"">Mod<sub><i>R</i></sub></a> with a principal ideal domain <i>R</i> is at most one.
"
"homotopy category (Wikipedia Glossary of Category Theory)"¬"
See <a href=""https://en.wikipedia.org/wiki/Homotopy_category"" title=""Homotopy category"">homotopy category</a>. It is closely related to a <a href=""https://en.wikipedia.org/wiki/Localization_of_a_category"" title=""Localization of a category"">localization of a category</a>.
"
"homotopy hypothesis (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Homotopy_hypothesis"" title=""Homotopy hypothesis"">homotopy hypothesis</a> states an <a href=""https://en.wikipedia.org/wiki/%E2%88%9E-groupoid"" title=""∞-groupoid"">∞-groupoid</a> is a space (less equivocally, an <i>n</i>-groupoid can be used as a homotopy <i>n</i>-type.)"
"identity (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;The <a href=""https://en.wikipedia.org/wiki/Identity_morphism"" class=""mw-redirect"" title=""Identity morphism"">identity morphism</a> <i>f</i> of an object <i>A</i> is a morphism from <i>A</i> to <i>A</i> such that for any morphisms <i>g</i> with domain <i>A</i> and <i>h</i> with codomain <i>A</i>, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle g\circ f=g}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>g</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
        <mo>=</mo>
        <mi>g</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle g\circ f=g}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/9bcac3b4321d1d8960b4aa35d19377410cbca08a"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:8.804ex; height:2.509ex;"" alt=""g\circ f=g""/></span> and <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f\circ h=h}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>h</mi>
        <mo>=</mo>
        <mi>h</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f\circ h=h}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/7371785d65317969d4719bbd3e1fad1d54a553b3"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:9.25ex; height:2.509ex;"" alt=""f\circ h=h""/></span>.</dd>
<dd class=""glossary"">2.&#160;&#160;The <a href=""https://en.wikipedia.org/wiki/Identity_functor"" class=""mw-redirect"" title=""Identity functor"">identity functor</a> on a category <i>C</i> is a functor from <i>C</i> to <i>C</i> that sends objects and morphisms to themselves.</dd>
<dd class=""glossary"">3.&#160;&#160;Given a functor <i>F</i>: <i>C</i> → <i>D</i>, the <a href=""https://en.wikipedia.org/wiki/Identity_natural_transformation"" class=""mw-redirect"" title=""Identity natural transformation"">identity natural transformation</a> from <i>F</i> to <i>F</i> is a natural transformation consisting of the identity morphisms of <i>F</i>(<i>X</i>) in <i>D</i> for the objects <i>X</i> in <i>C</i>.</dd>
"
"image (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Image_of_a_morphism"" class=""mw-redirect"" title=""Image of a morphism"">image</a> of a morphism <i>f</i>: <i>X</i> → <i>Y</i> is the equalizer of <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle Y\rightrightarrows Y\sqcup _{X}Y}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Y</mi>
        <mo stretchy=""false"">&#x21C9;<!-- ⇉ --></mo>
        <mi>Y</mi>
        <msub>
          <mo>&#x2294;<!-- ⊔ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mi>Y</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle Y\rightrightarrows Y\sqcup _{X}Y}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f59cc423b4b8d4658365473ffbe550063a17d5b0"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:13.149ex; height:2.509ex;"" alt=""{\displaystyle Y\rightrightarrows Y\sqcup _{X}Y}""/></span>.
"
"ind-limit (Wikipedia Glossary of Category Theory)"¬"
A colimit (or inductive limit) in <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \mathbf {Fct} (C^{\text{op}},\mathbf {Set} )}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">F</mi>
          <mi mathvariant=""bold"">c</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <msup>
          <mi>C</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mtext>op</mtext>
          </mrow>
        </msup>
        <mo>,</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \mathbf {Fct} (C^{\text{op}},\mathbf {Set} )}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/41677376119fdc5e92d1b122bb92ffb189c94228"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:14.269ex; height:2.843ex;"" alt=""\mathbf {Fct} (C^{\text{op}},\mathbf {Set} )""/></span>.
"
"inductive limit (Wikipedia Glossary of Category Theory)"¬"
Another name for <a href=""https://en.wikipedia.org/wiki/Colimit"" class=""mw-redirect"" title=""Colimit"">colimit</a>.
"
"∞-category (Wikipedia Glossary of Category Theory)"¬"
An <a href=""https://en.wikipedia.org/wiki/%E2%88%9E-category"" class=""mw-redirect"" title=""∞-category"">∞-category</a> <i>C</i> is a <a href=""https://en.wikipedia.org/wiki/Simplicial_set"" title=""Simplicial set"">simplicial set</a> satisfying the following condition: for each 0 &lt; <i>i</i> &lt; <i>n</i>,
<ul><li>every map of simplicial sets <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:\Lambda _{i}^{n}\to C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <msubsup>
          <mi mathvariant=""normal"">&#x039B;<!-- Λ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msubsup>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:\Lambda _{i}^{n}\to C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/5e5eb3ba09ba23380b1a96641760de0375a1ec7a"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.005ex; width:11.428ex; height:2.843ex;"" alt=""f:\Lambda _{i}^{n}\to C""/></span> extends to an <i>n</i>-simplex <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:\Delta ^{n}\to C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <msup>
          <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msup>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:\Delta ^{n}\to C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/3390c389a178d62500cf0af0fcb0703c2d711714"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:11.751ex; height:2.676ex;"" alt=""f:\Delta ^{n}\to C""/></span></li></ul>
where &#916;<sup><i>n</i></sup> is the standard <i>n</i>-simplex and <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \Lambda _{i}^{n}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msubsup>
          <mi mathvariant=""normal"">&#x039B;<!-- Λ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msubsup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \Lambda _{i}^{n}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/e83ace72faae848b80156cd2412a563ba8a7b0d2"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.005ex; width:2.832ex; height:2.843ex;"" alt=""\Lambda _{i}^{n}""/></span> is obtained from &#916;<sup><i>n</i></sup> by removing the <i>i</i>-th face and the interior (see <a href=""https://en.wikipedia.org/wiki/Kan_fibration#Definition"" title=""Kan fibration"">Kan fibration#Definition</a>). For example, the <a href=""https://en.wikipedia.org/wiki/Nerve_of_a_category"" class=""mw-redirect"" title=""Nerve of a category"">nerve of a category</a> satisfies the condition and thus can be considered as an ∞-category.
"
"initial (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;An object <i>A</i> is <a href=""https://en.wikipedia.org/wiki/Initial_object"" class=""mw-redirect"" title=""Initial object"">initial</a> if there is exactly one morphism from <i>A</i> to each object; e.g., <a href=""https://en.wikipedia.org/wiki/Empty_set"" title=""Empty set"">empty set</a> in <b><a href=""https://en.wikipedia.org/wiki/Category_of_sets"" title=""Category of sets"">Set</a></b>.</dd>
<dd class=""glossary"">2.&#160;&#160;An object <i>A</i> in an ∞-category <i>C</i> is initial if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Map} _{C}(A,B)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>Map</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>C</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>A</mi>
        <mo>,</mo>
        <mi>B</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Map} _{C}(A,B)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/88632c60099f646c43f42831070f3468005f9cbd"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:12.418ex; height:2.843ex;"" alt=""\operatorname {Map} _{C}(A,B)""/></span> is <a href=""https://en.wikipedia.org/wiki/Contractible_space"" title=""Contractible space"">contractible</a> for each object <i>B</i> in <i>C</i>.</dd>
"
"injective (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;An object <i>A</i> in an abelian category is <a href=""https://en.wikipedia.org/wiki/Injective_object"" title=""Injective object"">injective</a> if the functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} (-,A)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo>,</mo>
        <mi>A</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} (-,A)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/457e7b81821e4854c980b5a6080a4ef066fcdd91"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:11.236ex; height:2.843ex;"" alt=""\operatorname {Hom} (-,A)""/></span> is exact. It is the dual of a projective object.</dd>
<dd class=""glossary"">2.&#160;&#160;The term “injective limit” is another name for a <a href=""https://en.wikipedia.org/wiki/Direct_limit"" title=""Direct limit"">direct limit</a>.</dd>
"
"internal Hom (Wikipedia Glossary of Category Theory)"¬"
Given a <a href=""https://en.wikipedia.org/wiki/Monoidal_category"" title=""Monoidal category"">monoidal category</a> (<i>C</i>, ⊗), the <a href=""https://en.wikipedia.org/wiki/Internal_Hom"" class=""mw-redirect"" title=""Internal Hom"">internal Hom</a> is a functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle [-,-]:C^{\text{op}}\times C\to C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo stretchy=""false"">[</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo>,</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo stretchy=""false"">]</mo>
        <mo>:</mo>
        <msup>
          <mi>C</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mtext>op</mtext>
          </mrow>
        </msup>
        <mo>&#x00D7;<!-- × --></mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle [-,-]:C^{\text{op}}\times C\to C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ed445fc38423e0a2d715258f5078af089b92a823"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:21.634ex; height:2.843ex;"" alt=""[-,-]:C^{\text{op}}\times C\to C""/></span> such that <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle [Y,-]}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo stretchy=""false"">[</mo>
        <mi>Y</mi>
        <mo>,</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo stretchy=""false"">]</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle [Y,-]}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/993653bd51d16acf94844d01562b26517ccffa06"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:5.909ex; height:2.843ex;"" alt=""[Y,-]""/></span> is the right adjoint to <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle -\otimes Y}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo>&#x2212;<!-- − --></mo>
        <mo>&#x2297;<!-- ⊗ --></mo>
        <mi>Y</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle -\otimes Y}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ea71f27c5242465f4185c7577c4333874b50c404"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.505ex; width:6.422ex; height:2.343ex;"" alt=""-\otimes Y""/></span> for each object <i>Y</i> in <i>C</i>. For example, the <a href=""https://en.wikipedia.org/wiki/Category_of_modules"" title=""Category of modules"">category of modules</a> over a commutative ring <i>R</i> has the internal Hom given as <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle [M,N]=\operatorname {Hom} _{R}(M,N)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo stretchy=""false"">[</mo>
        <mi>M</mi>
        <mo>,</mo>
        <mi>N</mi>
        <mo stretchy=""false"">]</mo>
        <mo>=</mo>
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>R</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>M</mi>
        <mo>,</mo>
        <mi>N</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle [M,N]=\operatorname {Hom} _{R}(M,N)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c244ef22174bc5796d25aa70faf83a2c05476"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:23.602ex; height:2.843ex;"" alt=""[M,N]=\operatorname {Hom} _{R}(M,N)""/></span>, the set of <i>R</i>-linear maps.
"
"inverse (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;A morphism <i>f</i> is an <a href=""https://en.wikipedia.org/wiki/Inverse_function"" title=""Inverse function"">inverse</a> to a morphism <i>g</i> if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle g\circ f}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>g</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle g\circ f}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:4.589ex; height:2.509ex;"" alt=""g\circ f""/></span> is defined and is equal to the identity morphism on the codomain of <i>g</i>, and <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f\circ g}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>g</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f\circ g}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:4.589ex; height:2.509ex;"" alt=""f\circ g""/></span> is defined and equal to the identity morphism on the domain of <i>g</i>. The inverse of <i>g</i> is unique and is denoted by <i>g</i><sup>−1</sup>. <i>f</i> is a left inverse to <i>g</i> if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f\circ g}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>g</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f\circ g}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:4.589ex; height:2.509ex;"" alt=""f\circ g""/></span> is defined and is equal to the identity morphism on the domain of <i>g</i>, and similarly for a right inverse.</dd>
<dd class=""glossary"">2.&#160;&#160;An <a href=""https://en.wikipedia.org/wiki/Inverse_limit"" title=""Inverse limit"">inverse limit</a> is the limit of an <a href=""https://en.wikipedia.org/wiki/Inverse_system"" class=""mw-redirect"" title=""Inverse system"">inverse system</a>.</dd>
"
"isomorphic (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;An object is <a href=""https://en.wikipedia.org/wiki/Isomorphic"" class=""mw-redirect"" title=""Isomorphic"">isomorphic</a> to another object if there is an isomorphism between them.</dd>
<dd class=""glossary"">2.&#160;&#160;A category is isomorphic to another category if there is an isomorphism between them.</dd>
"
"isomorphism (Wikipedia Glossary of Category Theory)"¬"
A morphism <i>f</i> is an <a href=""https://en.wikipedia.org/wiki/Isomorphism"" title=""Isomorphism"">isomorphism</a> if there exists an <i>inverse</i> of <i>f</i>."
"Kan complex (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Kan_complex"" class=""mw-redirect"" title=""Kan complex"">Kan complex</a> is a <a href=""https://en.wikipedia.org/wiki/Fibrant_object"" title=""Fibrant object"">fibrant object</a> in the category of simplicial sets.
"
"Kan extension (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;Given a category <i>C</i>, the left <a href=""https://en.wikipedia.org/wiki/Kan_extension"" title=""Kan extension"">Kan extension</a> functor along a functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:I\to J}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>I</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>J</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:I\to J}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/23881bc96b93801b6109ee99c71a9f1bb7289de7"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:9.473ex; height:2.509ex;"" alt=""f: I \to J""/></span> is the left adjoint (if it exists) to <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f^{*}=-\circ f:\operatorname {Fct} (J,C)\to \operatorname {Fct} (I,C)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>f</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>&#x2217;<!-- ∗ --></mo>
          </mrow>
        </msup>
        <mo>=</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
        <mo>:</mo>
        <mi>Fct</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>J</mi>
        <mo>,</mo>
        <mi>C</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>Fct</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>I</mi>
        <mo>,</mo>
        <mi>C</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f^{*}=-\circ f:\operatorname {Fct} (J,C)\to \operatorname {Fct} (I,C)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/d9a7c5e8868b60e1701aeb69934000d836281426"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:35.078ex; height:2.843ex;"" alt=""{\displaystyle f^{*}=-\circ f:\operatorname {Fct} (J,C)\to \operatorname {Fct} (I,C)}""/></span> and is denoted by <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f_{!}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>f</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>!</mo>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f_{!}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/28b2946320392521c0aa3d0b60d6d873adf0d10d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:1.829ex; height:2.509ex;"" alt=""f_{!}""/></span>. For any <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \alpha :I\to C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>&#x03B1;<!-- α --></mi>
        <mo>:</mo>
        <mi>I</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \alpha :I\to C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d44e97a66d66d0015d5d9c9071b7fabee929ae"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:9.977ex; height:2.176ex;"" alt=""\alpha: I \to C""/></span>, the functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f_{!}\alpha :J\to C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>f</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>!</mo>
          </mrow>
        </msub>
        <mi>&#x03B1;<!-- α --></mi>
        <mo>:</mo>
        <mi>J</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f_{!}\alpha :J\to C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/3a35875087b08ce10778251131bffa0f10c821a3"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:12.105ex; height:2.509ex;"" alt=""f_! \alpha: J \to C""/></span> is called the left Kan extension of α along <i>f</i>.<sup id=""cite_ref-7"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-7"">&#91;7&#93;</a></sup> One can show:
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle (f_{!}\alpha )(j)=\varinjlim _{f(i)\to j}\alpha (i)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>f</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>!</mo>
          </mrow>
        </msub>
        <mi>&#x03B1;<!-- α --></mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">(</mo>
        <mi>j</mi>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <munder>
          <mrow class=""MJX-TeXAtom-OP"">
            <munder>
              <mi>lim</mi>
              <mo>&#x2192;<!-- → --></mo>
            </munder>
          </mrow>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>f</mi>
            <mo stretchy=""false"">(</mo>
            <mi>i</mi>
            <mo stretchy=""false"">)</mo>
            <mo stretchy=""false"">&#x2192;<!-- → --></mo>
            <mi>j</mi>
          </mrow>
        </munder>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>&#x03B1;<!-- α --></mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle (f_{!}\alpha )(j)=\varinjlim _{f(i)\to j}\alpha (i)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c1e6e7303aba3b1c734e8a822851159ba12e94"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -4.171ex; width:20.55ex; height:6.176ex;"" alt=""{\displaystyle (f_{!}\alpha )(j)=\varinjlim _{f(i)\to j}\alpha (i)}""/></span></dd></dl>
where the colimit runs over all objects <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f(i)\to j}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>j</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f(i)\to j}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/352a31ad65131e14940ee3c851502e810b122039"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:8.462ex; height:2.843ex;"" alt=""{\displaystyle f(i)\to j}""/></span> in the comma category.</dd>
<dd class=""glossary"">2.&#160;&#160;The right Kan extension functor is the right adjoint (if it exists) to <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f^{*}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>f</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>&#x2217;<!-- ∗ --></mo>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f^{*}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/190a73fde235865b8d2a783334f90194331c7f19"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:2.375ex; height:2.676ex;"" alt=""f^{*}""/></span>.</dd>
"
"Ken Brown's lemma (Wikipedia Glossary of Category Theory)"¬"
<a href=""https://en.wikipedia.org/w/index.php?title=Ken_Brown%27s_lemma&amp;action=edit&amp;redlink=1"" class=""new"" title=""Ken Brown&#39;s lemma (page does not exist)"">Ken Brown's lemma</a> is a lemma in the theory of model categories.
"
"Kleisli category (Wikipedia Glossary of Category Theory)"¬"
Given a monad <i>T</i>, the <a href=""https://en.wikipedia.org/wiki/Kleisli_category"" title=""Kleisli category"">Kleisli category</a> of <i>T</i> is the full subcategory of the category of <i>T</i>-algebras (called Eilenberg–Moore category) that consists of free <i>T</i>-algebras."
"lax (Wikipedia Glossary of Category Theory)"¬"
The term ""<a href=""https://en.wikipedia.org/wiki/Lax_functor"" title=""Lax functor"">lax functor</a>"" is essentially synonymous with ""<a href=""https://en.wikipedia.org/wiki/Pseudo-functor"" title=""Pseudo-functor"">pseudo-functor</a>"".
"
"length (Wikipedia Glossary of Category Theory)"¬"
An object in an abelian category is said to have <span id=""finite_length""></span>finite length if it has a <a href=""https://en.wikipedia.org/wiki/Composition_series"" title=""Composition series"">composition series</a>. The maximum number of proper subobjects in any such composition series is called the <b>length</b> of <i>A</i>.<sup id=""cite_ref-8"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-8"">&#91;8&#93;</a></sup>
"
"limit (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;The <a href=""https://en.wikipedia.org/wiki/Limit_(category_theory)"" title=""Limit (category theory)"">limit</a> (or <a href=""https://en.wikipedia.org/wiki/Projective_limit"" class=""mw-redirect"" title=""Projective limit"">projective limit</a>) of a functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:I^{\text{op}}\to \mathbf {Set} }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <msup>
          <mi>I</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mtext>op</mtext>
          </mrow>
        </msup>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:I^{\text{op}}\to \mathbf {Set} }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f0ec5d8573fe1c352064d0b28fe626998d5bc7a7"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:13.764ex; height:2.676ex;"" alt=""f:I^{\text{op}}\to \mathbf {Set} ""/></span> is
<dl><dd><dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varprojlim _{i\in I}f(i)=\{(x_{i}|i)\in \prod _{i}f(i)|f(s)(x_{j})=x_{i}{\text{ for any }}s:i\to j\}.}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <munder>
          <mrow class=""MJX-TeXAtom-OP"">
            <munder>
              <mi>lim</mi>
              <mo>&#x2190;<!-- ← --></mo>
            </munder>
          </mrow>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
            <mo>&#x2208;<!-- ∈ --></mo>
            <mi>I</mi>
          </mrow>
        </munder>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mo fence=""false"" stretchy=""false"">{</mo>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>x</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mo stretchy=""false"">|</mo>
        </mrow>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2208;<!-- ∈ --></mo>
        <munder>
          <mo>&#x220F;<!-- ∏ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </munder>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mo stretchy=""false"">|</mo>
        </mrow>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>s</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>x</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>j</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <msub>
          <mi>x</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mtext>&#xA0;for any&#xA0;</mtext>
        </mrow>
        <mi>s</mi>
        <mo>:</mo>
        <mi>i</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>j</mi>
        <mo fence=""false"" stretchy=""false"">}</mo>
        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varprojlim _{i\in I}f(i)=\{(x_{i}|i)\in \prod _{i}f(i)|f(s)(x_{j})=x_{i}{\text{ for any }}s:i\to j\}.}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/dd6824459027630ac2b3fb87613f3514de6eaaaf"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -3.671ex; width:59.962ex; height:6.176ex;"" alt=""\varprojlim _{i\in I}f(i)=\{(x_{i}|i)\in \prod _{i}f(i)|f(s)(x_{j})=x_{i}{\text{ for any }}s:i\to j\}.""/></span></dd></dl></dd></dl></dd></dl>
<dd class=""glossary"">2.&#160;&#160;The limit <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varprojlim _{i\in I}f(i)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <munder>
          <mrow class=""MJX-TeXAtom-OP"">
            <munder>
              <mi>lim</mi>
              <mo>&#x2190;<!-- ← --></mo>
            </munder>
          </mrow>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
            <mo>&#x2208;<!-- ∈ --></mo>
            <mi>I</mi>
          </mrow>
        </munder>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varprojlim _{i\in I}f(i)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5c676acef71aabcabf12d7678dbed001da4375"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -3.671ex; width:7.702ex; height:5.676ex;"" alt=""\varprojlim _{i\in I}f(i)""/></span> of a functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:I^{\text{op}}\to C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <msup>
          <mi>I</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mtext>op</mtext>
          </mrow>
        </msup>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:I^{\text{op}}\to C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/fa997fdf80892807a7bcde521a5fd033eb81e6db"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:11.781ex; height:2.676ex;"" alt=""f:I^{\text{op}}\to C""/></span> is an object, if any, in <i>C</i> that satisfies: for any object <i>X</i> in <i>C</i>, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} (X,\varprojlim _{i\in I}f(i))=\varprojlim _{i\in I}\operatorname {Hom} (X,f(i))}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <munder>
          <mrow class=""MJX-TeXAtom-OP"">
            <munder>
              <mi>lim</mi>
              <mo>&#x2190;<!-- ← --></mo>
            </munder>
          </mrow>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
            <mo>&#x2208;<!-- ∈ --></mo>
            <mi>I</mi>
          </mrow>
        </munder>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <munder>
          <mrow class=""MJX-TeXAtom-OP"">
            <munder>
              <mi>lim</mi>
              <mo>&#x2190;<!-- ← --></mo>
            </munder>
          </mrow>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
            <mo>&#x2208;<!-- ∈ --></mo>
            <mi>I</mi>
          </mrow>
        </munder>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} (X,\varprojlim _{i\in I}f(i))=\varprojlim _{i\in I}\operatorname {Hom} (X,f(i))}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/3202a82a1d3f240ad8ef642ff9a045b24ee44f41"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -3.671ex; width:37.832ex; height:5.676ex;"" alt=""\operatorname {Hom} (X,\varprojlim _{i\in I}f(i))=\varprojlim _{i\in I}\operatorname {Hom} (X,f(i))""/></span>; i.e., it is an object representing the functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle X\mapsto \varprojlim _{i}\operatorname {Hom} (X,f(i)).}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>X</mi>
        <mo stretchy=""false"">&#x21A6;<!-- ↦ --></mo>
        <munder>
          <mrow class=""MJX-TeXAtom-OP"">
            <munder>
              <mi>lim</mi>
              <mo>&#x2190;<!-- ← --></mo>
            </munder>
          </mrow>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </munder>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle X\mapsto \varprojlim _{i}\operatorname {Hom} (X,f(i)).}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/114b89599715a3fa7ac0acbed32aa7a7ee53655d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -3.671ex; width:23.608ex; height:5.676ex;"" alt=""X\mapsto \varprojlim _{i}\operatorname {Hom} (X,f(i)).""/></span></dd>
<dd class=""glossary"">3.&#160;&#160;The <a href=""https://en.wikipedia.org/wiki/Colimit"" class=""mw-redirect"" title=""Colimit"">colimit</a> (or <a href=""https://en.wikipedia.org/wiki/Inductive_limit"" class=""mw-redirect"" title=""Inductive limit"">inductive limit</a>) <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varinjlim _{i\in I}f(i)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <munder>
          <mrow class=""MJX-TeXAtom-OP"">
            <munder>
              <mi>lim</mi>
              <mo>&#x2192;<!-- → --></mo>
            </munder>
          </mrow>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
            <mo>&#x2208;<!-- ∈ --></mo>
            <mi>I</mi>
          </mrow>
        </munder>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varinjlim _{i\in I}f(i)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/a7ce63b06decfc6cf9c26448cc26b2abacea5c66"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -3.671ex; width:7.637ex; height:5.676ex;"" alt=""\varinjlim _{i\in I}f(i)""/></span> is the dual of a limit; i.e., given a functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:I\to C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>I</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:I\to C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/9271e0d23650020d206dcdb2277513962308597f"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:9.768ex; height:2.509ex;"" alt=""f:I\to C""/></span>, it satisfies: for any <i>X</i>, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} (\varinjlim f(i),X)=\varprojlim \operatorname {Hom} (f(i),X)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mrow class=""MJX-TeXAtom-OP"">
          <munder>
            <mi>lim</mi>
            <mo>&#x2192;<!-- → --></mo>
          </munder>
        </mrow>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mrow class=""MJX-TeXAtom-OP"">
          <munder>
            <mi>lim</mi>
            <mo>&#x2190;<!-- ← --></mo>
          </munder>
        </mrow>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} (\varinjlim f(i),X)=\varprojlim \operatorname {Hom} (f(i),X)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/092e5745ab1d81aca504fd0e0917bdcf1bd44dc4"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.637ex; margin-bottom: -0.367ex; width:37.767ex; height:4.009ex;"" alt=""\operatorname {Hom} (\varinjlim f(i),X)=\varprojlim \operatorname {Hom} (f(i),X)""/></span>. Explicitly, to give <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varinjlim f(i)\to X}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-OP"">
          <munder>
            <mi>lim</mi>
            <mo>&#x2192;<!-- → --></mo>
          </munder>
        </mrow>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varinjlim f(i)\to X}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9c784d4ef4959662491faa455460ed93bfb180"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.637ex; margin-bottom: -0.367ex; width:13.231ex; height:4.009ex;"" alt=""\varinjlim f(i)\to X""/></span> is to give a family of morphisms <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f(i)\to X}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f(i)\to X}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/e6098d7f6bd34bb55b9c3574e8cb4ef7b60738ce"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:9.484ex; height:2.843ex;"" alt=""f(i)\to X""/></span> such that for any <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle i\to j}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>i</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>j</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle i\to j}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/a48bd9ca3d9a2aa1b63faba78b32761d44301ddd"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:5.375ex; height:2.509ex;"" alt=""i\to j""/></span>, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f(i)\to X}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f(i)\to X}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/e6098d7f6bd34bb55b9c3574e8cb4ef7b60738ce"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:9.484ex; height:2.843ex;"" alt=""f(i)\to X""/></span> is <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f(i)\to f(j)\to X}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>j</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f(i)\to f(j)\to X}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/de5ed8aee0dacbbbc222f59fc49d8fb61bae80c8"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:17.144ex; height:2.843ex;"" alt=""f(i)\to f(j)\to X""/></span>. Perhaps the simplest example of a colimit is a <a href=""https://en.wikipedia.org/wiki/Coequalizer"" title=""Coequalizer"">coequalizer</a>. For another example, take <i>f</i> to be the identity functor on <i>C</i> and suppose <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle L=\varinjlim _{X\in C}f(X)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>L</mi>
        <mo>=</mo>
        <munder>
          <mrow class=""MJX-TeXAtom-OP"">
            <munder>
              <mi>lim</mi>
              <mo>&#x2192;<!-- → --></mo>
            </munder>
          </mrow>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
            <mo>&#x2208;<!-- ∈ --></mo>
            <mi>C</mi>
          </mrow>
        </munder>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle L=\varinjlim _{X\in C}f(X)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/aab689e6ab10080f17a120afc52e4c8113215f6d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -3.671ex; width:13.882ex; height:5.676ex;"" alt=""L=\varinjlim _{X\in C}f(X)""/></span> exists; then the identity morphism on <i>L</i> corresponds to a compatible family of morphisms <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \alpha _{X}:X\to L}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>&#x03B1;<!-- α --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mo>:</mo>
        <mi>X</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>L</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \alpha _{X}:X\to L}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f974ddf199a5d513fc630c7b761396713b31e8eb"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:12.234ex; height:2.509ex;"" alt=""\alpha _{X}:X\to L""/></span> such that <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \alpha _{L}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>&#x03B1;<!-- α --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>L</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \alpha _{L}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/5799c989016980850f6c191453e78b241c894323"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:2.839ex; height:2.009ex;"" alt=""\alpha _{L}""/></span> is the identity. If <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:X\to L}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>X</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>L</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:X\to L}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/830d77bcd269d327959be1e15fc5bd2282f9882f"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:10.393ex; height:2.509ex;"" alt=""f:X\to L""/></span> is any morphism, then <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f=\alpha _{L}\circ f=\alpha _{X}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>=</mo>
        <msub>
          <mi>&#x03B1;<!-- α --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>L</mi>
          </mrow>
        </msub>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
        <mo>=</mo>
        <msub>
          <mi>&#x03B1;<!-- α --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f=\alpha _{L}\circ f=\alpha _{X}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/21edd2e08a458921941b99ed1e20fbbce70d9a41"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:16.908ex; height:2.509ex;"" alt=""f=\alpha _{L}\circ f=\alpha _{X}""/></span>; i.e., <i>L</i> is a final object of <i>C</i>.
</dd>
"
"localization of a category (Wikipedia Glossary of Category Theory)"¬"
See <a href=""https://en.wikipedia.org/wiki/Localization_of_a_category"" title=""Localization of a category"">localization of a category</a>.
"
"Mittag-Leffler condition (Wikipedia Glossary of Category Theory)"¬"
An <a href=""https://en.wikipedia.org/wiki/Inverse_system"" class=""mw-redirect"" title=""Inverse system"">inverse system</a> <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \cdots \to X_{2}\to X_{1}\to X_{0}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo>&#x22EF;<!-- ⋯ --></mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>2</mn>
          </mrow>
        </msub>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>0</mn>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \cdots \to X_{2}\to X_{1}\to X_{0}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/327784bd585873faa4debf74c2c12f57e7e9acd2"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:22.501ex; height:2.509ex;"" alt=""{\displaystyle \cdots \to X_{2}\to X_{1}\to X_{0}}""/></span> is said to satisfy the <a href=""https://en.wikipedia.org/wiki/Mittag-Leffler_condition"" class=""mw-redirect"" title=""Mittag-Leffler condition"">Mittag-Leffler condition</a> if for each integer <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle n\geq 0}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>n</mi>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mn>0</mn>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle n\geq 0}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.505ex; width:5.656ex; height:2.343ex;"" alt=""n\geq 0""/></span>, there is an integer <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle m\geq n}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>m</mi>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mi>n</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle m\geq n}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/6b02f25e62da7fe3162ac80446437cdc1c0fd341"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.505ex; width:6.534ex; height:2.176ex;"" alt=""m\geq n""/></span> such that for each <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle l\geq m}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>l</mi>
        <mo>&#x2265;<!-- ≥ --></mo>
        <mi>m</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle l\geq m}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/affe2f50eafebdadd8b989319bd85df686e6bfea"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.505ex; width:5.832ex; height:2.343ex;"" alt=""{\displaystyle l\geq m}""/></span>, the images of <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle X_{m}\to X_{n}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>m</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle X_{m}\to X_{n}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/c1277bb3c16d916af52838fa89232360645d9c23"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:10.356ex; height:2.509ex;"" alt=""{\displaystyle X_{m}\to X_{n}}""/></span> and <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle X_{l}\to X_{n}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>l</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle X_{l}\to X_{n}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f1da23816a3b68986049441c66526ce5a499b1ec"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:9.404ex; height:2.509ex;"" alt=""{\displaystyle X_{l}\to X_{n}}""/></span> are the same.

"
"monad (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Monad_(category_theory)"" title=""Monad (category theory)"">monad</a> in a category <i>X</i> is a <a href=""https://en.wikipedia.org/wiki/Monoid_object"" class=""mw-redirect"" title=""Monoid object"">monoid object</a> in the monoidal category of endofunctors of <i>X</i> with the monoidal structure given by composition. For example, given a group <i>G</i>, define an endofunctor <i>T</i> on <b>Set</b> by <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle T(X)=G\times X}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>T</mi>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mi>G</mi>
        <mo>&#x00D7;<!-- × --></mo>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle T(X)=G\times X}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/079886d159c04dcacdb8bea94d74f7ceb267a57f"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:15.171ex; height:2.843ex;"" alt=""{\displaystyle T(X)=G\times X}""/></span>. Then define the multiplication <i>μ</i> on <i>T</i> as the natural transformation <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \mu :T\circ T\to T}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>&#x03BC;<!-- μ --></mi>
        <mo>:</mo>
        <mi>T</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>T</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>T</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \mu :T\circ T\to T}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ab127261db352655e7cdc155ed3691a571efb0d3"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:14.056ex; height:2.676ex;"" alt=""{\displaystyle \mu :T\circ T\to T}""/></span> given by
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \mu _{X}:G\times (G\times X)\to G\times X,\,\,(g,(h,x))\mapsto (gh,x)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>&#x03BC;<!-- μ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mo>:</mo>
        <mi>G</mi>
        <mo>&#x00D7;<!-- × --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>G</mi>
        <mo>&#x00D7;<!-- × --></mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>G</mi>
        <mo>&#x00D7;<!-- × --></mo>
        <mi>X</mi>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mspace width=""thinmathspace"" />
        <mo stretchy=""false"">(</mo>
        <mi>g</mi>
        <mo>,</mo>
        <mo stretchy=""false"">(</mo>
        <mi>h</mi>
        <mo>,</mo>
        <mi>x</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x21A6;<!-- ↦ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>g</mi>
        <mi>h</mi>
        <mo>,</mo>
        <mi>x</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \mu _{X}:G\times (G\times X)\to G\times X,\,\,(g,(h,x))\mapsto (gh,x)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/874024c405ef14fbbac7f842002ee2d806200185"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:49.877ex; height:2.843ex;"" alt=""{\displaystyle \mu _{X}:G\times (G\times X)\to G\times X,\,\,(g,(h,x))\mapsto (gh,x)}""/></span></dd></dl>
and also define the identity map <i>η</i> in the analogous fashion. Then (<i>T</i>, <i>μ</i>, <i>η</i>) constitutes a monad in <b>Set</b>. More substantially, an adjunction between functors <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F:X\rightleftarrows A:G}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <mo>:</mo>
        <mi>X</mi>
        <mo stretchy=""false"">&#x21C4;<!-- ⇄ --></mo>
        <mi>A</mi>
        <mo>:</mo>
        <mi>G</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F:X\rightleftarrows A:G}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/b528debf5f87ec4d3db742cba5cb2c2f58da59e3"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:14.779ex; height:2.176ex;"" alt=""{\displaystyle F:X\rightleftarrows A:G}""/></span> determines a monad in <i>X</i>; namely, one takes <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle T=G\circ F}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>T</mi>
        <mo>=</mo>
        <mi>G</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>F</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle T=G\circ F}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/658aaf5bcf5d379b85de489e141031b419636b80"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:10.497ex; height:2.176ex;"" alt=""T=G\circ F""/></span>, the identity map <i>η</i> on <i>T</i> to be a unit of the adjunction and also defines <i>μ</i> using the adjunction.
"
"monadic (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;An adjunction is said to be <a href=""https://en.wikipedia.org/wiki/Monadic_adjunction"" class=""mw-redirect"" title=""Monadic adjunction"">monadic</a> if it comes from the monad that it determines by means of the <a href=""https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Moore_category"" class=""mw-redirect"" title=""Eilenberg–Moore category"">Eilenberg–Moore category</a> (the category of algebras for the monad).</dd>
<dd class=""glossary"">2.&#160;&#160;A functor is said to be <a href=""https://en.wikipedia.org/wiki/Monadic_functor"" class=""mw-redirect"" title=""Monadic functor"">monadic</a> if it is a constituent of a monadic adjunction.</dd>
"
"monoidal category (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Monoidal_category"" title=""Monoidal category"">monoidal category</a>, also called a tensor category, is a category <i>C</i> equipped with (1) a <a href=""https://en.wikipedia.org/wiki/Bifunctor"" class=""mw-redirect"" title=""Bifunctor"">bifunctor</a> <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \otimes :C\times C\to C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo>&#x2297;<!-- ⊗ --></mo>
        <mo>:</mo>
        <mi>C</mi>
        <mo>&#x00D7;<!-- × --></mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \otimes :C\times C\to C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/d28e24da78ac0e7abe5ffffa235241c097ab5d2b"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.505ex; width:15.499ex; height:2.343ex;"" alt=""{\displaystyle \otimes :C\times C\to C}""/></span>, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.
"
"monoid object (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Monoid_object"" class=""mw-redirect"" title=""Monoid object"">monoid object</a> in a monoidal category is an object together with the multiplication map and the identity map that satisfy the expected conditions like associativity. For example, a monoid object in <b>Set</b> is a usual monoid (unital semigroup) and a monoid object in <b><a href=""https://en.wikipedia.org/wiki/Category_of_modules"" title=""Category of modules""><i>R</i>-mod</a></b> is an <a href=""https://en.wikipedia.org/wiki/Associative_algebra"" title=""Associative algebra"">associative algebra</a> over a commutative ring <i>R</i>.
"
"monomorphism (Wikipedia Glossary of Category Theory)"¬"
A morphism <i>f</i> is a <a href=""https://en.wikipedia.org/wiki/Monomorphism"" title=""Monomorphism"">monomorphism</a> (also called monic) if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle g=h}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>g</mi>
        <mo>=</mo>
        <mi>h</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle g=h}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/50f4fd70b1200ffb96f9b7dccc5b64296e8da9f4"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:5.553ex; height:2.509ex;"" alt=""g=h""/></span> whenever <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f\circ g=f\circ h}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>g</mi>
        <mo>=</mo>
        <mi>f</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>h</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f\circ g=f\circ h}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/aced36b06004c1fe84cee7ee4647d6d0e3e0e442"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:12.5ex; height:2.509ex;"" alt=""f\circ g=f\circ h""/></span>; e.g., an <a href=""https://en.wikipedia.org/wiki/Injective_function"" title=""Injective function"">injection</a> in <b><a href=""https://en.wikipedia.org/wiki/Category_of_sets"" title=""Category of sets"">Set</a></b>. In other words, <i>f</i> is the dual of an epimorphism.
"
"multicategory (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Multicategory"" title=""Multicategory"">multicategory</a> is a generalization of a category in which a morphism is allowed to have more than one domain. It is the same thing as a <a href=""https://en.wikipedia.org/wiki/Colored_operad"" class=""mw-redirect"" title=""Colored operad"">colored operad</a>.<sup id=""cite_ref-9"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-9"">&#91;9&#93;</a></sup>
"
"<i>n</i>-category (Wikipedia Glossary of Category Theory)"¬"
<div class=""quotebox pullquote floatright"" style=""width:33%;&#32;;""><style data-mw-deduplicate=""TemplateStyles:r887775652"">.mw-parser-output .quotebox{background-color:#F9F9F9;border:1px solid #aaa;box-sizing:border-box;padding:10px;font-size:88%;max-width:100%}.mw-parser-output .quotebox.floatleft{margin:0.5em 1.4em 0.8em 0}.mw-parser-output .quotebox.floatright{margin:0.5em 0 0.8em 1.4em}.mw-parser-output .quotebox.centered{margin:0.5em auto 0.8em auto}.mw-parser-output .quotebox.floatleft p,.mw-parser-output .quotebox.floatright p{font-style:inherit}.mw-parser-output .quotebox-title{background-color:#F9F9F9;text-align:center;font-size:larger;font-weight:bold}.mw-parser-output .quotebox-quote.quoted:before{font-family:""Times New Roman"",serif;font-weight:bold;font-size:large;color:gray;content:"" “ "";vertical-align:-45%;line-height:0}.mw-parser-output .quotebox-quote.quoted:after{font-family:""Times New Roman"",serif;font-weight:bold;font-size:large;color:gray;content:"" ” "";line-height:0}.mw-parser-output .quotebox .left-aligned{text-align:left}.mw-parser-output .quotebox .right-aligned{text-align:right}.mw-parser-output .quotebox .center-aligned{text-align:center}.mw-parser-output .quotebox cite{display:block;font-style:normal}@media screen and (max-width:360px){.mw-parser-output .quotebox{min-width:100%;margin:0 0 0.8em!important;float:none!important}}</style>
<div class=""quotebox-quote left-aligned"" style="""">[T]he issue of comparing definitions of weak <i>n</i>-category is a slippery one, as it is hard to say what it even <i>means</i> for two such definitions to be equivalent. [...] It is widely held that the structure formed by weak <i>n</i>-categories and the functors, transformations, ... between them should be a weak (<i>n</i> + 1)-category; and if this is the case then the question is whether your weak (<i>n</i> + 1)-category of weak <i>n-</i>categories is equivalent to mine—but whose definition of weak (<i>n</i> + 1)-category are we using here...&#160;?</div>
<p><cite class=""left-aligned"" style="""">Tom Leinster,&#32;<a rel=""nofollow"" class=""external text"" href=""http://www.tac.mta.ca/tac/volumes/10/1/10-01abs.html"">A survey of definitions of <i>n</i>-category</a></cite>
</p>
</div><dd class=""glossary"">1.&#160;&#160;A <a href=""https://en.wikipedia.org/wiki/Strict_n-category"" class=""mw-redirect"" title=""Strict n-category"">strict <i>n</i>-category</a> is defined inductively: a strict 0-category is a set and a strict <i>n</i>-category is a category whose Hom sets are strict (<i>n</i>-1)-categories. Precisely, a strict <i>n</i>-category is a category enriched over strict (<i>n</i>-1)-categories. For example, a strict 1-category is an ordinary category.</dd>
<dd class=""glossary"">2.&#160;&#160;The notion of a <a href=""https://en.wikipedia.org/wiki/Weak_n-category"" title=""Weak n-category"">weak <i>n</i>-category</a> is obtained from the strict one by weakening the conditions like associativity of composition to hold only up to <a href=""https://en.wikipedia.org/wiki/Coherent_isomorphism"" class=""mw-redirect"" title=""Coherent isomorphism"">coherent isomorphisms</a> in the weak sense.</dd>
<dd class=""glossary"">3.&#160;&#160;One can define an ∞-category as a kind of a colim of <i>n</i>-categories. Conversely, if one has the notion of a (weak) ∞-category (say a <a href=""https://en.wikipedia.org/wiki/Quasi-category"" title=""Quasi-category"">quasi-category</a>) in the beginning, then a weak <i>n</i>-category can be defined as a type of a truncated ∞-category.</dd>
"
"natural (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;A natural transformation is, roughly, a map between functors. Precisely, given a pair of functors <i>F</i>, <i>G</i> from a category <i>C</i> to category <i>D</i>, a <a href=""https://en.wikipedia.org/wiki/Natural_transformation"" title=""Natural transformation"">natural transformation</a> φ from <i>F</i> to <i>G</i> is a set of morphisms in <i>D</i>
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \{\phi _{x}:F(x)\to G(x)\mid x\in \operatorname {Ob} (C)\}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo fence=""false"" stretchy=""false"">{</mo>
        <msub>
          <mi>&#x03D5;<!-- ϕ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>x</mi>
          </mrow>
        </msub>
        <mo>:</mo>
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>x</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>G</mi>
        <mo stretchy=""false"">(</mo>
        <mi>x</mi>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2223;<!-- ∣ --></mo>
        <mi>x</mi>
        <mo>&#x2208;<!-- ∈ --></mo>
        <mi>Ob</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>C</mi>
        <mo stretchy=""false"">)</mo>
        <mo fence=""false"" stretchy=""false"">}</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \{\phi _{x}:F(x)\to G(x)\mid x\in \operatorname {Ob} (C)\}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/da61c25dbb0472cd348dcfd1ee6f80206e551ab5"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:33.063ex; height:2.843ex;"" alt=""{\displaystyle \{\phi _{x}:F(x)\to G(x)\mid x\in \operatorname {Ob} (C)\}}""/></span></dd></dl>
satisfying the condition: for each morphism <i>f</i>: <i>x</i> → <i>y</i> in <i>C</i>, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \phi _{y}\circ F(f)=G(f)\circ \phi _{x}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>&#x03D5;<!-- ϕ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>y</mi>
          </mrow>
        </msub>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mi>G</mi>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2218;<!-- ∘ --></mo>
        <msub>
          <mi>&#x03D5;<!-- ϕ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>x</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \phi _{y}\circ F(f)=G(f)\circ \phi _{x}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/60ec6093d1998a801af4db2ea8a5acec1868608c"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.005ex; width:22.224ex; height:3.009ex;"" alt=""\phi _{y}\circ F(f)=G(f)\circ \phi _{x}""/></span>. For example, writing <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle GL_{n}(R)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>G</mi>
        <msub>
          <mi>L</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">(</mo>
        <mi>R</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle GL_{n}(R)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/7772003961f3b3638965b7ecff4a2a9dade4a596"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:8.201ex; height:2.843ex;"" alt=""GL_{n}(R)""/></span> for the group of invertible <i>n</i>-by-<i>n</i> matrices with coefficients in a commutative ring <i>R</i>, we can view <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle GL_{n}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>G</mi>
        <msub>
          <mi>L</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle GL_{n}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/a41d6b4f9b104c2e8f659319ddbd61792518ace2"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:4.628ex; height:2.509ex;"" alt=""GL_{n}""/></span> as a functor from the category <b>CRing</b> of commutative rings to the category <b>Grp</b> of groups. Similarly, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle R\mapsto R^{*}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>R</mi>
        <mo stretchy=""false"">&#x21A6;<!-- ↦ --></mo>
        <msup>
          <mi>R</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>&#x2217;<!-- ∗ --></mo>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle R\mapsto R^{*}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/aa453fb5c6fa1b55cd4e68ac4ec4f98de6274868"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:8.196ex; height:2.343ex;"" alt=""R\mapsto R^{*}""/></span> is a functor from <b>CRing</b> to <b>Grp</b>. Then the <a href=""https://en.wikipedia.org/wiki/Determinant"" title=""Determinant"">determinant</a> det is a natural transformation from <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle GL_{n}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>G</mi>
        <msub>
          <mi>L</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle GL_{n}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/a41d6b4f9b104c2e8f659319ddbd61792518ace2"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:4.628ex; height:2.509ex;"" alt=""GL_{n}""/></span> to -<sup>*</sup>.</dd>
<dd class=""glossary"">2.&#160;&#160;A <a href=""https://en.wikipedia.org/wiki/Natural_isomorphism"" class=""mw-redirect"" title=""Natural isomorphism"">natural isomorphism</a> is a natural transformation that is an isomorphism (i.e., admits the inverse).</dd>

<div class=""thumb tright""><div class=""thumbinner"" style=""width:95px;""><a href=""https://en.wikipedia.org/wiki/File:Nerve-2-simplex.png"" class=""image""><img alt="""" src=""//upload.wikimedia.org/wikipedia/commons/d/d4/Nerve-2-simplex.png"" decoding=""async"" width=""93"" height=""98"" class=""thumbimage"" data-file-width=""93"" data-file-height=""98"" /></a>  <div class=""thumbcaption""><div class=""magnify""><a href=""https://en.wikipedia.org/wiki/File:Nerve-2-simplex.png"" class=""internal"" title=""Enlarge""></a></div>The composition is encoded as a 2-simplex.</div></div></div>"
"nerve (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Nerve_functor"" class=""mw-redirect"" title=""Nerve functor"">nerve functor</a> <i>N</i> is the functor from <b>Cat</b> to <i>s</i><b>Set</b> given by <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle N(C)_{n}=\operatorname {Hom} _{\mathbf {Cat} }([n],C)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>N</mi>
        <mo stretchy=""false"">(</mo>
        <mi>C</mi>
        <msub>
          <mo stretchy=""false"">)</mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mrow class=""MJX-TeXAtom-ORD"">
              <mi mathvariant=""bold"">C</mi>
              <mi mathvariant=""bold"">a</mi>
              <mi mathvariant=""bold"">t</mi>
            </mrow>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mo stretchy=""false"">[</mo>
        <mi>n</mi>
        <mo stretchy=""false"">]</mo>
        <mo>,</mo>
        <mi>C</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle N(C)_{n}=\operatorname {Hom} _{\mathbf {Cat} }([n],C)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc5e4e9263501b3a9614032ada2409f406e842d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:25.347ex; height:2.843ex;"" alt=""N(C)_{n}=\operatorname {Hom} _{\mathbf {Cat} }([n],C)""/></span>. For example, if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varphi }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>&#x03C6;<!-- φ --></mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varphi }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:1.52ex; height:2.176ex;"" alt=""\varphi ""/></span> is a functor in <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle N(C)_{2}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>N</mi>
        <mo stretchy=""false"">(</mo>
        <mi>C</mi>
        <msub>
          <mo stretchy=""false"">)</mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>2</mn>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle N(C)_{2}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/8ea73d5c5380ab85d836192a139e09337803f18b"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:6.693ex; height:2.843ex;"" alt=""{\displaystyle N(C)_{2}}""/></span> (called a 2-simplex), let <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle x_{i}=\varphi (i),\,0\leq i\leq 2}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>x</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mi>&#x03C6;<!-- φ --></mi>
        <mo stretchy=""false"">(</mo>
        <mi>i</mi>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mn>0</mn>
        <mo>&#x2264;<!-- ≤ --></mo>
        <mi>i</mi>
        <mo>&#x2264;<!-- ≤ --></mo>
        <mn>2</mn>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle x_{i}=\varphi (i),\,0\leq i\leq 2}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/938e78f4b1fb03bb1f6be4b3ec6a9d54d8b722cb"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:20.105ex; height:2.843ex;"" alt=""{\displaystyle x_{i}=\varphi (i),\,0\leq i\leq 2}""/></span>. Then <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varphi (0\to 1)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>&#x03C6;<!-- φ --></mi>
        <mo stretchy=""false"">(</mo>
        <mn>0</mn>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mn>1</mn>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varphi (0\to 1)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/c763ace1f7f103be54baf566a4e55cfc2bf4a9e5"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:9.268ex; height:2.843ex;"" alt=""{\displaystyle \varphi (0\to 1)}""/></span> is a morphism <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:x_{0}\to x_{1}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <msub>
          <mi>x</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>0</mn>
          </mrow>
        </msub>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msub>
          <mi>x</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>1</mn>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:x_{0}\to x_{1}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/c9451757da910036fc5f85497dc3cf2311939022"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:11.598ex; height:2.509ex;"" alt=""{\displaystyle f:x_{0}\to x_{1}}""/></span> in <i>C</i> and also <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varphi (1\to 2)=g:x_{1}\to x_{2}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>&#x03C6;<!-- φ --></mi>
        <mo stretchy=""false"">(</mo>
        <mn>1</mn>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mn>2</mn>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mi>g</mi>
        <mo>:</mo>
        <msub>
          <mi>x</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msub>
          <mi>x</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>2</mn>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varphi (1\to 2)=g:x_{1}\to x_{2}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/5784e3864c55d2166d498a979f2c158715a40e1a"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:23.802ex; height:2.843ex;"" alt=""{\displaystyle \varphi (1\to 2)=g:x_{1}\to x_{2}}""/></span> for some <i>g</i> in <i>C</i>. Since <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle 0\to 2}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mn>0</mn>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mn>2</mn>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle 0\to 2}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ea16222a69197d920f62ad0b69ffa4adf48cbe09"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:5.939ex; height:2.176ex;"" alt=""{\displaystyle 0\to 2}""/></span> is <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle 0\to 1}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mn>0</mn>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mn>1</mn>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle 0\to 1}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/385148d2752cc8c677f4c0b31f4ff0f5cad05303"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:5.939ex; height:2.176ex;"" alt=""{\displaystyle 0\to 1}""/></span> followed by <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle 1\to 2}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mn>1</mn>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mn>2</mn>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle 1\to 2}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1bd8612b838faf69959d07a28067b87f47577f"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:5.939ex; height:2.176ex;"" alt=""{\displaystyle 1\to 2}""/></span> and since <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varphi }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>&#x03C6;<!-- φ --></mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varphi }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:1.52ex; height:2.176ex;"" alt=""\varphi ""/></span> is a functor, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varphi (0\to 2)=g\circ f}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>&#x03C6;<!-- φ --></mi>
        <mo stretchy=""false"">(</mo>
        <mn>0</mn>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mn>2</mn>
        <mo stretchy=""false"">)</mo>
        <mo>=</mo>
        <mi>g</mi>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varphi (0\to 2)=g\circ f}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba559e213735c1ec67ee5ace3992a1f11fb86c6"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:16.956ex; height:2.843ex;"" alt=""{\displaystyle \varphi (0\to 2)=g\circ f}""/></span>. In other words, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varphi }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>&#x03C6;<!-- φ --></mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varphi }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:1.52ex; height:2.176ex;"" alt=""\varphi ""/></span> encodes <i>f</i>, <i>g</i> and their compositions.
"
"normal (Wikipedia Glossary of Category Theory)"¬"
A monomorphism is normal if it is the kernel of some morphism, and an epimorphism is conormal if it is the cokernel of some morphism.  A category is <a href=""https://en.wikipedia.org/wiki/Normal_category"" class=""mw-redirect"" title=""Normal category"">normal</a> if every monomorphism is normal."
"object (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;An object is part of a data defining a category.</dd>
<dd class=""glossary"">2.&#160;&#160;An [adjective] object in a category <i>C</i> is a contravariant functor (or presheaf) from some fixed category corresponding to the ""adjective"" to <i>C</i>. For example, a <a href=""https://en.wikipedia.org/wiki/Simplicial_object"" class=""mw-redirect"" title=""Simplicial object"">simplicial object</a> in <i>C</i> is a contravariant functor from the simplicial category to <i>C</i> and a <a href=""https://en.wikipedia.org/w/index.php?title=%CE%93-object&amp;action=edit&amp;redlink=1"" class=""new"" title=""Γ-object (page does not exist)"">Γ-object</a> is a pointed contravariant functor from <a href=""https://en.wikipedia.org/w/index.php?title=%CE%93_(category_theory)&amp;action=edit&amp;redlink=1"" class=""new"" title=""Γ (category theory) (page does not exist)"">Γ</a> (roughly the pointed category of pointed finite sets) to <i>C</i> provided <i>C</i> is pointed.</dd>
"
"op-fibration (Wikipedia Glossary of Category Theory)"¬"
A functor π:<i>C</i> → <i>D</i> is an <a href=""https://en.wikipedia.org/wiki/Op-fibration"" class=""mw-redirect"" title=""Op-fibration"">op-fibration</a> if, for each object <i>x</i> in <i>C</i> and each morphism <i>g</i>&#160;: π(<i>x</i>) → <i>y</i> in <i>D</i>, there is at least one π-coCartesian morphism <i>f</i>: <i>x</i> → <i>y'</i> in <i>C</i> such that π(<i>f</i>) = <i>g</i>. In other words, π is the dual of a <a href=""https://en.wikipedia.org/wiki/Grothendieck_fibration"" class=""mw-redirect"" title=""Grothendieck fibration"">Grothendieck fibration</a>.
"
"opposite (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Opposite_category"" title=""Opposite category"">opposite category</a> of a category is obtained by reversing the arrows. For example, if a partially ordered set is viewed as a category, taking its opposite amounts to reversing the ordering."
"perfect (Wikipedia Glossary of Category Theory)"¬"
Sometimes synonymous with ""compact"". See <a href=""https://en.wikipedia.org/wiki/Perfect_complex"" title=""Perfect complex"">perfect complex</a>.
"
"pointed (Wikipedia Glossary of Category Theory)"¬"
A category (or ∞-category) is called pointed if it has a zero object.
"
"polynomial (Wikipedia Glossary of Category Theory)"¬"
A functor from the category of finite-dimensional vector spaces to itself is called a <a href=""https://en.wikipedia.org/wiki/Polynomial_functor"" title=""Polynomial functor"">polynomial functor</a> if, for each pair of vector spaces <i>V</i>, <i>W</i>, <span class=""nowrap""><i>F</i>: Hom(<i>V</i>, <i>W</i>) → Hom(<i>F</i>(<i>V</i>), <i>F</i>(<i>W</i>))</span> is a polynomial map between the vector spaces. A <a href=""https://en.wikipedia.org/wiki/Schur_functor"" title=""Schur functor"">Schur functor</a> is a basic example.
"
"preadditive (Wikipedia Glossary of Category Theory)"¬"
A category is <a href=""https://en.wikipedia.org/wiki/Preadditive_category"" title=""Preadditive category"">preadditive</a> if it is <a href=""https://en.wikipedia.org/wiki/Enriched_category"" title=""Enriched category"">enriched</a> over the <a href=""https://en.wikipedia.org/wiki/Monoidal_category"" title=""Monoidal category"">monoidal category</a> of <a href=""https://en.wikipedia.org/wiki/Abelian_group"" title=""Abelian group"">abelian groups</a>. More generally, it is <a href=""https://en.wikipedia.org/wiki/Preadditive_category#R-linear_categories"" title=""Preadditive category""><i>R</i>-linear</a> if it is enriched over the monoidal category of <a href=""https://en.wikipedia.org/wiki/Module_(mathematics)"" title=""Module (mathematics)""><i>R</i>-modules</a>, for <i>R</i> a <a href=""https://en.wikipedia.org/wiki/Commutative_ring"" title=""Commutative ring"">commutative ring</a>.
"
"presentable (Wikipedia Glossary of Category Theory)"¬"
Given a <a href=""https://en.wikipedia.org/wiki/Regular_cardinal"" title=""Regular cardinal"">regular cardinal</a> κ, a category is <a href=""https://en.wikipedia.org/w/index.php?title=Presentabl_category&amp;action=edit&amp;redlink=1"" class=""new"" title=""Presentabl category (page does not exist)"">κ-presentable</a> if it admits all small colimits and is <a href=""#accessible"">κ-accessible</a>. A category is presentable if it is κ-presentable for some regular cardinal κ (hence presentable for any larger cardinal). <b>Note</b>: Some authors call a presentable category a <a href=""https://en.wikipedia.org/wiki/Locally_presentable_category"" class=""mw-redirect"" title=""Locally presentable category"">locally presentable category</a>.
"
"presheaf (Wikipedia Glossary of Category Theory)"¬"
Another term for a contravariant functor: a functor from a category <i>C</i><sup>op</sup> to <b>Set</b> is a presheaf of sets on <i>C</i> and a functor from <i>C</i><sup>op</sup> to <i>s</i><b>Set</b> is a presheaf of simplicial sets or <a href=""https://en.wikipedia.org/wiki/Simplicial_presheaf"" title=""Simplicial presheaf"">simplicial presheaf</a>, etc. A <a href=""https://en.wikipedia.org/wiki/Grothendieck_topology"" title=""Grothendieck topology"">topology</a> on <i>C</i>, if any, tells which presheaf is a sheaf (with respect to that topology).
"
"product (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;The <a href=""https://en.wikipedia.org/wiki/Product_(category_theory)"" title=""Product (category theory)"">product</a> of a family of objects <i>X</i><sub><i>i</i></sub> in a category <i>C</i> indexed by a set <i>I</i> is the projective limit <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \varprojlim }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mrow class=""MJX-TeXAtom-OP"">
          <munder>
            <mi>lim</mi>
            <mo>&#x2190;<!-- ← --></mo>
          </munder>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \varprojlim }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f742f0ca71dad3a357440193bd488b321f965c74"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -1.637ex; margin-bottom: -0.367ex; width:3.425ex; height:3.843ex;"" alt=""\varprojlim ""/></span> of the functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle I\to C,\,i\mapsto X_{i}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>I</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mi>i</mi>
        <mo stretchy=""false"">&#x21A6;<!-- ↦ --></mo>
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle I\to C,\,i\mapsto X_{i}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/a1df6b6f6ee5f004136520fcd3742d2911f2a4fb"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:15.114ex; height:2.509ex;"" alt=""I\to C,\,i\mapsto X_{i}""/></span>, where <i>I</i> is viewed as a discrete category. It is denoted by <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \prod _{i}X_{i}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <munder>
          <mo>&#x220F;<!-- ∏ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </munder>
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \prod _{i}X_{i}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/25951f7a02818dda8ba0c0a0224df04506bc6694"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -3.005ex; width:6.08ex; height:5.509ex;"" alt=""\prod _{i}X_{i}""/></span> and is the dual of the coproduct of the family.</dd>
<dd class=""glossary"">2.&#160;&#160;The <a href=""https://en.wikipedia.org/wiki/Product_of_categories"" class=""mw-redirect"" title=""Product of categories"">product of a family of categories</a> <i>C</i><sub><i>i</i></sub>'s indexed by a set <i>I</i> is the category denoted by <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \prod _{i}C_{i}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <munder>
          <mo>&#x220F;<!-- ∏ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </munder>
        <msub>
          <mi>C</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \prod _{i}C_{i}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/1a1908e8b4d789a8fa987f888b63f513c895a371"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -3.005ex; width:5.818ex; height:5.509ex;"" alt=""\prod _{i}C_{i}""/></span> whose class of objects is the product of the classes of objects of <i>C</i><sub><i>i</i></sub>'s and whose hom-sets are <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \prod _{i}\operatorname {Hom} _{\operatorname {C_{i}} }(X_{i},Y_{i})}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <munder>
          <mo>&#x220F;<!-- ∏ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </munder>
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mrow class=""MJX-TeXAtom-OP MJX-fixedlimits"">
              <msub>
                <mi mathvariant=""normal"">C</mi>
                <mrow class=""MJX-TeXAtom-ORD"">
                  <mi mathvariant=""normal"">i</mi>
                </mrow>
              </msub>
            </mrow>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
        <mo>,</mo>
        <msub>
          <mi>Y</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>i</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \prod _{i}\operatorname {Hom} _{\operatorname {C_{i}} }(X_{i},Y_{i})}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c4ea1990cd6406ade2373811b639a5e1bdc0e2"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -3.005ex; width:17.87ex; height:5.509ex;"" alt=""\prod _{i}\operatorname {Hom} _{\operatorname {C_{i}} }(X_{i},Y_{i})""/></span>; the morphisms are composed component-wise. It is the dual of the disjoint union.</dd>
"
"profunctor (Wikipedia Glossary of Category Theory)"¬"
Given categories <i>C</i> and <i>D</i>, a <a href=""https://en.wikipedia.org/wiki/Profunctor"" title=""Profunctor"">profunctor</a> (or a distributor) from <i>C</i> to <i>D</i> is a functor of the form <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle D^{\text{op}}\times C\to \mathbf {Set} }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>D</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mtext>op</mtext>
          </mrow>
        </msup>
        <mo>&#x00D7;<!-- × --></mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle D^{\text{op}}\times C\to \mathbf {Set} }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c5c81ba3986db2d9df90976554191ad32db811"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:15.863ex; height:2.343ex;"" alt=""{\displaystyle D^{\text{op}}\times C\to \mathbf {Set} }""/></span>.
"
"projective (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;An object <i>A</i> in an abelian category is <a href=""https://en.wikipedia.org/wiki/Projective_object"" title=""Projective object"">projective</a> if the functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} (A,-)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>A</mi>
        <mo>,</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} (A,-)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd09b0dd4114a6f799460830a49b59cb7c3d543"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:11.236ex; height:2.843ex;"" alt=""\operatorname {Hom} (A,-)""/></span> is exact. It is the dual of an injective object.</dd>
<dd class=""glossary"">2.&#160;&#160;The term “projective limit” is another name for an <a href=""https://en.wikipedia.org/wiki/Inverse_limit"" title=""Inverse limit"">inverse limit</a>.</dd>
"
"PROP (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/PROP_(category_theory)"" title=""PROP (category theory)"">PROP</a> is a symmetric strict monoidal category whose objects are natural numbers and whose tensor product <a href=""https://en.wikipedia.org/wiki/Addition"" title=""Addition"">addition</a> of natural numbers.
"
"pseudoalgebra (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Pseudoalgebra"" title=""Pseudoalgebra"">pseudoalgebra</a> is a 2-category-version of an algebra for a monad (with a monad replaced by a 2-monad)."
"Quillen (Wikipedia Glossary of Category Theory)"¬"
<a href=""https://en.wikipedia.org/wiki/Quillen%E2%80%99s_theorem_A"" class=""mw-redirect"" title=""Quillen’s theorem A"">Quillen’s theorem A</a> provides a criterion for a functor to be a weak equivalence."
"reflect (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;A functor is said to reflect identities if it has the property: if <i>F</i>(<i>k</i>) is an identity then <i>k</i> is an identity as well.</dd>
<dd class=""glossary"">2.&#160;&#160;A functor is said to reflect isomorphisms if it has the property:  <i>F</i>(<i>k</i>) is an isomorphism then <i>k</i> is an isomorphism as well.</dd>
"
"representable (Wikipedia Glossary of Category Theory)"¬"
A set-valued contravariant functor <i>F</i> on a category <i>C</i> is said to be <a href=""https://en.wikipedia.org/wiki/Representable_functor"" title=""Representable functor"">representable</a> if it belongs to the essential image of the <a href=""https://en.wikipedia.org/wiki/Yoneda_embedding"" class=""mw-redirect"" title=""Yoneda embedding"">Yoneda embedding</a> <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle C\to \mathbf {Fct} (C^{\text{op}},\mathbf {Set} )}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">F</mi>
          <mi mathvariant=""bold"">c</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <msup>
          <mi>C</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mtext>op</mtext>
          </mrow>
        </msup>
        <mo>,</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle C\to \mathbf {Fct} (C^{\text{op}},\mathbf {Set} )}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd3744e7835e41071e5af18cb2a01c6570b7ee6"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:19.65ex; height:2.843ex;"" alt=""C\to \mathbf {Fct} (C^{\text{op}},\mathbf {Set} )""/></span>; i.e., <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F\simeq \operatorname {Hom} _{C}(-,Z)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <mo>&#x2243;<!-- ≃ --></mo>
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>C</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo>,</mo>
        <mi>Z</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F\simeq \operatorname {Hom} _{C}(-,Z)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/2448bd47df48dd229837a2657c63b34b1629d6f8"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:17.494ex; height:2.843ex;"" alt=""F\simeq \operatorname {Hom} _{C}(-,Z)""/></span> for some object <i>Z</i>. The object <i>Z</i> is said to be the representing object of <i>F</i>.
"
"retraction (Wikipedia Glossary of Category Theory)"¬"
<div class=""thumb tright""><div class=""thumbinner"" style=""width:152px;""><a href=""https://en.wikipedia.org/wiki/File:Section_retract.svg"" class=""image""><img alt="""" src=""//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Section_retract.svg/150px-Section_retract.svg.png"" decoding=""async"" width=""150"" height=""166"" class=""thumbimage"" srcset=""//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Section_retract.svg/225px-Section_retract.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Section_retract.svg/300px-Section_retract.svg.png 2x"" data-file-width=""113"" data-file-height=""125"" /></a>  <div class=""thumbcaption""><div class=""magnify""><a href=""https://en.wikipedia.org/wiki/File:Section_retract.svg"" class=""internal"" title=""Enlarge""></a></div><i>f</i> is a retraction of <i>g</i>. <i>g</i> is a section of <i>f</i>.</div></div></div>A morphism is a <a href=""https://en.wikipedia.org/wiki/Section_(category_theory)"" title=""Section (category theory)"">retraction</a> if it has a right inverse."
"section (Wikipedia Glossary of Category Theory)"¬"
A morphism is a <a href=""https://en.wikipedia.org/wiki/Section_(category_theory)"" title=""Section (category theory)"">section</a> if it has a left inverse. For example, the <a href=""https://en.wikipedia.org/wiki/Axiom_of_choice"" title=""Axiom of choice"">axiom of choice</a> says that any surjective function admits a section.
"
"Segal space (Wikipedia Glossary of Category Theory)"¬"
<a href=""https://en.wikipedia.org/wiki/Segal_space"" title=""Segal space"">Segal spaces</a> were certain simplicial spaces, introduced as models for <a href=""https://en.wikipedia.org/wiki/(infinity,1)-category"" class=""mw-redirect"" title=""(infinity,1)-category"">(∞,&#160;1)-categories</a>.
"
"semisimple (Wikipedia Glossary of Category Theory)"¬"
An abelian category is <a href=""https://en.wikipedia.org/wiki/Semisimple_category"" class=""mw-redirect"" title=""Semisimple category"">semisimple</a> if every short exact sequence splits. For example, a ring is <a href=""https://en.wikipedia.org/wiki/Semisimple_ring"" class=""mw-redirect"" title=""Semisimple ring"">semisimple</a> if and only if the category of modules over it is semisimple.
"
"Serre functor (Wikipedia Glossary of Category Theory)"¬"
Given a <i>k</i>-linear category <i>C</i> over a field <i>k</i>, a <a href=""https://en.wikipedia.org/w/index.php?title=Serre_functor&amp;action=edit&amp;redlink=1"" class=""new"" title=""Serre functor (page does not exist)"">Serre functor</a> <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:C\to C}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>C</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:C\to C}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/c995236c557387fa3c4a70e23b85d6375b597108"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:10.362ex; height:2.509ex;"" alt=""f:C\to C""/></span> is an auto-equivalence such that <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} (A,B)\simeq \operatorname {Hom} (B,f(A))^{*}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>A</mi>
        <mo>,</mo>
        <mi>B</mi>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2243;<!-- ≃ --></mo>
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>B</mi>
        <mo>,</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>A</mi>
        <mo stretchy=""false"">)</mo>
        <msup>
          <mo stretchy=""false"">)</mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>&#x2217;<!-- ∗ --></mo>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} (A,B)\simeq \operatorname {Hom} (B,f(A))^{*}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8d2eadb1040bd65ad6b70453b805bbcc4dc9a9"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:29.624ex; height:2.843ex;"" alt=""{\displaystyle \operatorname {Hom} (A,B)\simeq \operatorname {Hom} (B,f(A))^{*}}""/></span> for any objects <i>A</i>, <i>B</i>."
"simple object (Wikipedia Glossary of Category Theory)"¬"
A simple object in an abelian category is <span id=""simple_object""></span>an object <i>A</i> that is not isomorphic to the zero object and whose every <a href=""https://en.wikipedia.org/wiki/Subobject"" title=""Subobject"">subobject</a> is isomorphic to zero or to <i>A</i>. For example, a <a href=""https://en.wikipedia.org/wiki/Simple_module"" title=""Simple module"">simple module</a> is precisely a simple object in the category of (say left) modules.
"
"simplex category (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Simplex_category"" title=""Simplex category"">simplex category</a> &#916; is the category where an object is a set [<i>n</i>] = { 0, 1, …, <i>n</i> }, n ≥ 0, totally ordered in the standard way and a morphism is an order-preserving function.
"
"simplicial category (Wikipedia Glossary of Category Theory)"¬"
A category enriched over simplicial sets.
"
"Simplicial localization (Wikipedia Glossary of Category Theory)"¬"
<a href=""https://en.wikipedia.org/wiki/Simplicial_localization"" title=""Simplicial localization"">Simplicial localization</a> is a method of localizing a category.
"
"simplicial object (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Simplicial_object"" class=""mw-redirect"" title=""Simplicial object"">simplicial object</a> in a category <i>C</i> is roughly a sequence of objects <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle X_{0},X_{1},X_{2},\dots }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>0</mn>
          </mrow>
        </msub>
        <mo>,</mo>
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>1</mn>
          </mrow>
        </msub>
        <mo>,</mo>
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mn>2</mn>
          </mrow>
        </msub>
        <mo>,</mo>
        <mo>&#x2026;<!-- … --></mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle X_{0},X_{1},X_{2},\dots }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/e068505f72a88bf2c295490331817273592e0788"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:14.761ex; height:2.509ex;"" alt=""{\displaystyle X_{0},X_{1},X_{2},\dots }""/></span> in <i>C</i> that forms a simplicial set. In other words, it is a covariant or contravariant functor Δ → <i>C</i>. For example, a <a href=""https://en.wikipedia.org/wiki/Simplicial_presheaf"" title=""Simplicial presheaf"">simplicial presheaf</a> is a simplicial object in the category of presheaves.
"
"simplicial set (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Simplicial_set"" title=""Simplicial set"">simplicial set</a> is a contravariant functor from &#916; to <b>Set</b>, where &#916; is the <a href=""https://en.wikipedia.org/wiki/Simplex_category"" title=""Simplex category"">simplex category</a>, a category whose objects are the sets [<i>n</i>] = { 0, 1, …, <i>n</i> } and whose morphisms are order-preserving functions. One writes <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle X_{n}=X([n])}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mi>X</mi>
        <mo stretchy=""false"">(</mo>
        <mo stretchy=""false"">[</mo>
        <mi>n</mi>
        <mo stretchy=""false"">]</mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle X_{n}=X([n])}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8558bc385bb78fbaed546ad8bca07d30418e8e"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:12.719ex; height:2.843ex;"" alt=""{\displaystyle X_{n}=X([n])}""/></span> and an element of the set <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle X_{n}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle X_{n}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:3.143ex; height:2.509ex;"" alt=""X_{n}""/></span> is called an <i>n</i>-simplex. For example, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \Delta ^{n}=\operatorname {Hom} _{\Delta }(-,[n])}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msup>
        <mo>=</mo>
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo>,</mo>
        <mo stretchy=""false"">[</mo>
        <mi>n</mi>
        <mo stretchy=""false"">]</mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \Delta ^{n}=\operatorname {Hom} _{\Delta }(-,[n])}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/2f897c7e6cc659851325cea5f722b1f8fc099323"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:20.035ex; height:2.843ex;"" alt=""{\displaystyle \Delta ^{n}=\operatorname {Hom} _{\Delta }(-,[n])}""/></span> is a simplicial set called the standard <i>n</i>-simplex. By Yoneda's lemma, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle X_{n}\simeq \operatorname {Nat} (\Delta ^{n},X)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>X</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msub>
        <mo>&#x2243;<!-- ≃ --></mo>
        <mi>Nat</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <msup>
          <mi mathvariant=""normal"">&#x0394;<!-- Δ --></mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>n</mi>
          </mrow>
        </msup>
        <mo>,</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle X_{n}\simeq \operatorname {Nat} (\Delta ^{n},X)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/7d06364bd3e2b3064b3d6c5cbd33a17fa725430a"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:18.029ex; height:2.843ex;"" alt=""{\displaystyle X_{n}\simeq \operatorname {Nat} (\Delta ^{n},X)}""/></span>.
"
"site (Wikipedia Glossary of Category Theory)"¬"
A category equipped with a <a href=""https://en.wikipedia.org/wiki/Grothendieck_topology"" title=""Grothendieck topology"">Grothendieck topology</a>.
"
"skeletal (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;A category is <a href=""https://en.wikipedia.org/wiki/Skeleton_(category_theory)"" title=""Skeleton (category theory)"">skeletal</a> if isomorphic objects are necessarily identical.</dd>
<dd class=""glossary"">2.&#160;&#160;A (not unique) <a href=""https://en.wikipedia.org/wiki/Skeleton_(category_theory)"" title=""Skeleton (category theory)"">skeleton</a> of a category is a full subcategory that is skeletal.</dd>
"
"slice (Wikipedia Glossary of Category Theory)"¬"
Given a category <i>C</i> and an object <i>A</i> in it, the <a href=""https://en.wikipedia.org/wiki/Slice_category"" class=""mw-redirect"" title=""Slice category"">slice category</a> <i>C</i><sub>/<i>A</i></sub> of <i>C</i> over <i>A</i> is the category whose objects are all the morphisms in <i>C</i> with codomain <i>A</i>, whose morphisms are morphisms in <i>C</i> such that if <i>f</i> is a morphism from <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle p_{X}:X\to A}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>p</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mo>:</mo>
        <mi>X</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>A</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle p_{X}:X\to A}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/19f64ff90ada2fa28838f564c9b2c1c87d9ae39e"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; margin-left: -0.089ex; width:12.166ex; height:2.509ex;"" alt=""p_{X}:X\to A""/></span> to <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle p_{Y}:Y\to A}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>p</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>Y</mi>
          </mrow>
        </msub>
        <mo>:</mo>
        <mi>Y</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>A</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle p_{Y}:Y\to A}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/a58834445b4217fb53df7a7b561421e722ef6b7c"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; margin-left: -0.089ex; width:11.813ex; height:2.509ex;"" alt=""p_{Y}:Y\to A""/></span>, then <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle p_{Y}\circ f=p_{X}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>p</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>Y</mi>
          </mrow>
        </msub>
        <mo>&#x2218;<!-- ∘ --></mo>
        <mi>f</mi>
        <mo>=</mo>
        <msub>
          <mi>p</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle p_{Y}\circ f=p_{X}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/13b01c104ec1bb3d88d3052babab31bd8ce6f8b3"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; margin-left: -0.089ex; width:12.119ex; height:2.509ex;"" alt=""p_{Y}\circ f=p_{X}""/></span> in <i>C</i> and whose composition is that of <i>C</i>.
"
"small (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;A <a href=""https://en.wikipedia.org/wiki/Small_category"" class=""mw-redirect"" title=""Small category"">small category</a> is a category in which the class of all morphisms is a <a href=""https://en.wikipedia.org/wiki/Set_(mathematics)"" title=""Set (mathematics)"">set</a> (i.e., not a <a href=""https://en.wikipedia.org/wiki/Proper_class"" class=""mw-redirect"" title=""Proper class"">proper class</a>); otherwise <b>large</b>. A category is <b>locally small</b> if the morphisms between every pair of objects <i>A</i> and <i>B</i> form a set. Some authors assume a foundation in which the collection of all classes forms a ""conglomerate"", in which case a <b>quasicategory</b> is a category whose objects and morphisms merely form a <a href=""https://en.wikipedia.org/wiki/Conglomerate_(mathematics)"" title=""Conglomerate (mathematics)"">conglomerate</a>.<sup id=""cite_ref-10"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-10"">&#91;10&#93;</a></sup> (NB: some authors use the term ""quasicategory"" with a different meaning.<sup id=""cite_ref-11"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-11"">&#91;11&#93;</a></sup>)</dd>
<dd class=""glossary"">2.&#160;&#160;An object in a category is said to be <a href=""https://en.wikipedia.org/w/index.php?title=Small_object&amp;action=edit&amp;redlink=1"" class=""new"" title=""Small object (page does not exist)"">small</a> if it is κ-compact for some regular cardinal κ. The notion prominently appears in Quiilen's <a href=""https://en.wikipedia.org/w/index.php?title=Small_object_argument&amp;action=edit&amp;redlink=1"" class=""new"" title=""Small object argument (page does not exist)"">small object argument</a> (cf. <a rel=""nofollow"" class=""external free"" href=""https://ncatlab.org/nlab/show/small+object+argument"">https://ncatlab.org/nlab/show/small+object+argument</a>)</dd>
"
"species (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Combinatorial_species"" title=""Combinatorial species"">(combinatorial) species</a> is an endofunctor on the groupoid of finite sets with bijections. It is categorically equivalent to a <a href=""https://en.wikipedia.org/wiki/Symmetric_sequence"" class=""mw-redirect"" title=""Symmetric sequence"">symmetric sequence</a>.
"
"stable (Wikipedia Glossary of Category Theory)"¬"
An ∞-category is <a href=""https://en.wikipedia.org/wiki/Stable_%E2%88%9E-category"" title=""Stable ∞-category"">stable</a> if (1) it has a zero object, (2) every morphism in it admits a fiber and a cofiber and (3) a triangle in it is a fiber sequence if and only if it is a cofiber sequence.
"
"strict (Wikipedia Glossary of Category Theory)"¬"
A morphism <i>f</i> in a category admitting finite limits and finite colimits is <a href=""https://en.wikipedia.org/w/index.php?title=Strict_morphism&amp;action=edit&amp;redlink=1"" class=""new"" title=""Strict morphism (page does not exist)"">strict</a> if the natural morphism <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Coim} (f)\to \operatorname {Im} (f)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Coim</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>Im</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Coim} (f)\to \operatorname {Im} (f)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd6f2101c22c928397f5217eacfe4ff85ef46cd"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:17.989ex; height:2.843ex;"" alt=""\operatorname {Coim} (f)\to \operatorname {Im} (f)""/></span> is an isomorphism.
"
"strict <i>n</i>-category (Wikipedia Glossary of Category Theory)"¬"
A strict 0-category is a set and for any integer <i>n</i> &gt; 0, a <a href=""https://en.wikipedia.org/wiki/Strict_n-category"" class=""mw-redirect"" title=""Strict n-category"">strict <i>n</i>-category</a> is a category enriched over strict (<i>n</i>-1)-categories. For example, a strict 1-category is an ordinary category. <b>Note</b>: the term ""<i>n</i>-category"" typically refers to ""<a href=""https://en.wikipedia.org/wiki/Weak_n-category"" title=""Weak n-category"">weak <i>n</i>-category</a>""; not strict one.
"
"subcanonical (Wikipedia Glossary of Category Theory)"¬"
A topology on a category is <a href=""https://en.wikipedia.org/w/index.php?title=Subcanonical&amp;action=edit&amp;redlink=1"" class=""new"" title=""Subcanonical (page does not exist)"">subcanonical</a> if every representable contravariant functor on <i>C</i> is a sheaf with respect to that topology.<sup id=""cite_ref-12"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-12"">&#91;12&#93;</a></sup> Generally speaking, some <a href=""https://en.wikipedia.org/wiki/Flat_topology"" title=""Flat topology"">flat topology</a> may fail to be subcanonical; but flat topologies appearing in practice tend to be subcanonical.
"
"subcategory (Wikipedia Glossary of Category Theory)"¬"
A category <i>A</i> is a <a href=""https://en.wikipedia.org/wiki/Subcategory"" title=""Subcategory"">subcategory</a> of a category <i>B</i> if there is an inclusion functor from <i>A</i> to <i>B</i>.
"
"subobject (Wikipedia Glossary of Category Theory)"¬"
Given an object <i>A</i> in a category, a <a href=""https://en.wikipedia.org/wiki/Subobject"" title=""Subobject"">subobject</a> of <i>A</i> is an equivalence class of monomorphisms to <i>A</i>; two monomorphisms <i>f</i>, <i>g</i> are considered equivalent if <i>f</i> factors through <i>g</i> and <i>g</i> factors through <i>f</i>.
"
"subquotient (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Subquotient"" title=""Subquotient"">subquotient</a> is a quotient of a subobject.
"
"subterminal object (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Subterminal_object"" title=""Subterminal object"">subterminal object</a> is an object <i>X</i> such that every object has at most one morphism into <i>X</i>.
"
"symmetric monoidal category (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Symmetric_monoidal_category"" title=""Symmetric monoidal category"">symmetric monoidal category</a> is a <a href=""https://en.wikipedia.org/wiki/Monoidal_category"" title=""Monoidal category"">monoidal category</a> (i.e., a category with ⊗) that has maximally symmetric braiding.
"
"symmetric sequence (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Symmetric_sequence"" class=""mw-redirect"" title=""Symmetric sequence"">symmetric sequence</a> is a sequence of objects with actions of <a href=""https://en.wikipedia.org/wiki/Symmetric_group"" title=""Symmetric group"">symmetric groups</a>. It is categorically equivalent to a <a href=""https://en.wikipedia.org/wiki/Combinatorial_species"" title=""Combinatorial species"">(combinatorial) species</a>.
"
"t-structure (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/T-structure"" title=""T-structure"">t-structure</a> is an additional structure on a <a href=""https://en.wikipedia.org/wiki/Triangulated_category"" title=""Triangulated category"">triangulated category</a> (more generally <a href=""https://en.wikipedia.org/wiki/Stable_%E2%88%9E-category"" title=""Stable ∞-category"">stable ∞-category</a>) that axiomatizes the notions of complexes whose cohomology concentrated in non-negative degrees or non-positive degrees.
"
"Tannakian duality (Wikipedia Glossary of Category Theory)"¬"
The <a href=""https://en.wikipedia.org/wiki/Tannakian_duality"" class=""mw-redirect"" title=""Tannakian duality"">Tannakian duality</a> states that, in an appropriate setup, to give a morphism <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:X\to Y}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>X</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>Y</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:X\to Y}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:10.583ex; height:2.509ex;"" alt=""f:X\to Y""/></span> is to give a pullback functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f^{*}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msup>
          <mi>f</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mo>&#x2217;<!-- ∗ --></mo>
          </mrow>
        </msup>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f^{*}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/190a73fde235865b8d2a783334f90194331c7f19"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:2.375ex; height:2.676ex;"" alt=""f^{*}""/></span> along it. In other words, the Hom set <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} (X,Y)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Hom</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>Y</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} (X,Y)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5ad87d5a22d6eadb0dafdabdd0f55237821860"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:11.438ex; height:2.843ex;"" alt=""\operatorname {Hom} (X,Y)""/></span> can be identified with the functor category <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Fct} (D(Y),D(X))}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>Fct</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>D</mi>
        <mo stretchy=""false"">(</mo>
        <mi>Y</mi>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mi>D</mi>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Fct} (D(Y),D(X))}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/346f664c595157d3d0ce232822d8457d310dddc5"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:17.519ex; height:2.843ex;"" alt=""{\displaystyle \operatorname {Fct} (D(Y),D(X))}""/></span>, perhaps in the <a href=""https://en.wikipedia.org/wiki/Derived_algebraic_geometry"" title=""Derived algebraic geometry"">derived sense</a>, where <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle D(X)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>D</mi>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle D(X)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/054ce5cebeedabbdd1656658239c5da5264c2b96"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:5.714ex; height:2.843ex;"" alt=""D(X)""/></span> is the category associated to <i>X</i> (e.g., the derived category).<sup id=""cite_ref-13"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-13"">&#91;13&#93;</a></sup><sup id=""cite_ref-14"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-14"">&#91;14&#93;</a></sup>
"
"tensor category (Wikipedia Glossary of Category Theory)"¬"
Usually synonymous with <a href=""https://en.wikipedia.org/wiki/Monoidal_category"" title=""Monoidal category"">monoidal category</a> (though some authors distinguish between the two concepts.)
"
"tensor triangulated category (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/w/index.php?title=Tensor_triangulated_category&amp;action=edit&amp;redlink=1"" class=""new"" title=""Tensor triangulated category (page does not exist)"">tensor triangulated category</a> is a category that carries the structure of a symmetric monoidal category and that of a triangulated category in a compatible way.
"
"tensor product (Wikipedia Glossary of Category Theory)"¬"
Given a monoidal category <i>B</i>, the <b>tensor product of functors</b> <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F:C^{\text{op}}\to B}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <mo>:</mo>
        <msup>
          <mi>C</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mtext>op</mtext>
          </mrow>
        </msup>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>B</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F:C^{\text{op}}\to B}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/63cfc37c7985672bd726565d38adcce69323dba7"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:12.822ex; height:2.343ex;"" alt=""F:C^{\text{op}}\to B""/></span> and <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle G:C\to B}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>G</mi>
        <mo>:</mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>B</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle G:C\to B}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/4a0e30b5bdba4e56b8177c4d963c081549afaf6d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:10.908ex; height:2.176ex;"" alt=""G:C\to B""/></span> is the coend:
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F\otimes _{C}G=\int ^{c\in C}F(c)\otimes G(c).}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <msub>
          <mo>&#x2297;<!-- ⊗ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>C</mi>
          </mrow>
        </msub>
        <mi>G</mi>
        <mo>=</mo>
        <msup>
          <mo>&#x222B;<!-- ∫ --></mo>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>c</mi>
            <mo>&#x2208;<!-- ∈ --></mo>
            <mi>C</mi>
          </mrow>
        </msup>
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2297;<!-- ⊗ --></mo>
        <mi>G</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F\otimes _{C}G=\int ^{c\in C}F(c)\otimes G(c).}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/50e3a3ecb57f1c5f583303875b900c0594c7b157"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -2.338ex; width:29.815ex; height:6.343ex;"" alt=""{\displaystyle F\otimes _{C}G=\int ^{c\in C}F(c)\otimes G(c).}""/></span></dd></dl></dl>
"
"terminal (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;An object <i>A</i> is <a href=""https://en.wikipedia.org/wiki/Terminal_object"" class=""mw-redirect"" title=""Terminal object"">terminal</a> (also called final) if there is exactly one morphism from each object to <i>A</i>; e.g., <a href=""https://en.wikipedia.org/wiki/Singleton_(mathematics)"" title=""Singleton (mathematics)"">singletons</a>  in <b><a href=""https://en.wikipedia.org/wiki/Category_of_sets"" title=""Category of sets"">Set</a></b>. It is the dual of an <a href=""https://en.wikipedia.org/wiki/Initial_object"" class=""mw-redirect"" title=""Initial object"">initial object</a>.</dd>
<dd class=""glossary"">2.&#160;&#160;An object <i>A</i> in an ∞-category <i>C</i> is terminal if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Map} _{C}(B,A)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>Map</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>C</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>B</mi>
        <mo>,</mo>
        <mi>A</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Map} _{C}(B,A)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/8a813c027b796c2ced18c4ce0cad72c92a2016d5"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:12.418ex; height:2.843ex;"" alt=""\operatorname {Map} _{C}(B,A)""/></span> is <a href=""https://en.wikipedia.org/wiki/Contractible_space"" title=""Contractible space"">contractible</a> for every object <i>B</i> in <i>C</i>.</dd>
"
"thick subcategory (Wikipedia Glossary of Category Theory)"¬"
A full subcategory of an abelian category is <a href=""https://en.wikipedia.org/w/index.php?title=Thick_subcategory&amp;action=edit&amp;redlink=1"" class=""new"" title=""Thick subcategory (page does not exist)"">thick</a> if it is closed under extensions.
"
"thin (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Thin_category"" class=""mw-redirect"" title=""Thin category"">thin</a> is a category where there is at most one morphism between any pair of objects.
"
"triangulated category (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Triangulated_category"" title=""Triangulated category"">triangulated category</a> is a category where one can talk about distinguished triangles, generalization of exact sequences. An abelian category is a prototypical example of a triangulated category. A <a href=""https://en.wikipedia.org/wiki/Derived_category"" title=""Derived category"">derived category</a> is a triangulated category that is not necessary an abelian category.
"
"universal (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;Given a functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle f:C\to D}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>f</mi>
        <mo>:</mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>D</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle f:C\to D}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/f9af673473fce78b42b08129374eef6a040b35bb"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:10.52ex; height:2.509ex;"" alt=""f:C\to D""/></span> and an object <i>X</i> in <i>D</i>, a <a href=""https://en.wikipedia.org/wiki/Universal_morphism"" class=""mw-redirect"" title=""Universal morphism"">universal morphism</a> from <i>X</i> to <i>f</i> is an initial object in the <a href=""https://en.wikipedia.org/wiki/Comma_category"" title=""Comma category"">comma category</a> <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle (X\downarrow f)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo stretchy=""false"">&#x2193;<!-- ↓ --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle (X\downarrow f)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/7027c2136ee1783c68a71f3443d994254dad81f6"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:7.521ex; height:2.843ex;"" alt=""(X\downarrow f)""/></span>. (Its dual is also called a universal morphism.) For example, take <i>f</i> to be the forgetful functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \mathbf {Vec} _{k}\to \mathbf {Set} }"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi mathvariant=""bold"">V</mi>
            <mi mathvariant=""bold"">e</mi>
            <mi mathvariant=""bold"">c</mi>
          </mrow>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>k</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \mathbf {Vec} _{k}\to \mathbf {Set} }</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/1349060c5fe60d72fa8ba37800512d8acdf22d1d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:12.885ex; height:2.509ex;"" alt=""\mathbf {Vec} _{k}\to \mathbf {Set} ""/></span> and <i>X</i> a set. An initial object of <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle (X\downarrow f)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo stretchy=""false"">&#x2193;<!-- ↓ --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle (X\downarrow f)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/7027c2136ee1783c68a71f3443d994254dad81f6"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:7.521ex; height:2.843ex;"" alt=""(X\downarrow f)""/></span> is a function <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle j:X\to f(V_{X})}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>j</mi>
        <mo>:</mo>
        <mi>X</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>V</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle j:X\to f(V_{X})}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/5d47a9463baf506a11f6ea0fafcc21c1614c98a1"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; margin-left: -0.027ex; width:14.591ex; height:2.843ex;"" alt=""j:X\to f(V_{X})""/></span>. That it is initial means that if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle k:X\to f(W)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>k</mi>
        <mo>:</mo>
        <mi>X</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>W</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle k:X\to f(W)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/01b2e1a98ad889631f8aac7993d7a6c1c0cb9890"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:14.266ex; height:2.843ex;"" alt=""k:X\to f(W)""/></span> is another morphism, then there is a unique morphism from <i>j</i> to <i>k</i>, which consists of a linear map <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle V_{X}\to W}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>V</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>W</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle V_{X}\to W}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/c64b585a4f904c1b63ab4e294f9b4e3b1f573979"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:9.037ex; height:2.509ex;"" alt=""V_{X}\to W""/></span> that extends <i>k</i> via <i>j</i>; that is to say, <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle V_{X}}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>V</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle V_{X}}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/b6aec7566ccba7a946eca2221504813eb0408903"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.671ex; width:2.988ex; height:2.509ex;"" alt=""V_{X}""/></span> is the <a href=""https://en.wikipedia.org/wiki/Free_vector_space"" class=""mw-redirect"" title=""Free vector space"">free vector space</a> generated by <i>X</i>.</dd>
<dd class=""glossary"">2.&#160;&#160;Stated more explicitly, given <i>f</i> as above, a morphism <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle X\to f(u_{X})}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>X</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>u</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle X\to f(u_{X})}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/aa57cd04e45b0583b44ef636df19d8ef32d1499e"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:11.644ex; height:2.843ex;"" alt=""X\to f(u_{X})""/></span> in <i>D</i> is universal if and only if the natural map
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} _{C}(u_{X},c)\to \operatorname {Hom} _{D}(X,f(c)),\,\alpha \mapsto (X\to f(u_{x}){\overset {f(\alpha )}{\to }}f(c))}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>C</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>u</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mo>,</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>D</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mi>&#x03B1;<!-- α --></mi>
        <mo stretchy=""false"">&#x21A6;<!-- ↦ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>u</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>x</mi>
          </mrow>
        </msub>
        <mo stretchy=""false"">)</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mover>
            <mo stretchy=""false"">&#x2192;<!-- → --></mo>
            <mrow>
              <mi>f</mi>
              <mo stretchy=""false"">(</mo>
              <mi>&#x03B1;<!-- α --></mi>
              <mo stretchy=""false"">)</mo>
            </mrow>
          </mover>
        </mrow>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mi>c</mi>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} _{C}(u_{X},c)\to \operatorname {Hom} _{D}(X,f(c)),\,\alpha \mapsto (X\to f(u_{x}){\overset {f(\alpha )}{\to }}f(c))}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/4ad2bafe6189aa1f71cf3e118b146e4ff75c6b84"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:58.947ex; height:4.343ex;"" alt=""\operatorname {Hom} _{C}(u_{X},c)\to \operatorname {Hom} _{D}(X,f(c)),\,\alpha \mapsto (X\to f(u_{x}){\overset {f(\alpha )}{\to }}f(c))""/></span></dd></dl>
is bijective. In particular, if <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} _{C}(u_{X},-)\simeq \operatorname {Hom} _{D}(X,f(-))}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>C</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>u</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>X</mi>
          </mrow>
        </msub>
        <mo>,</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2243;<!-- ≃ --></mo>
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>D</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} _{C}(u_{X},-)\simeq \operatorname {Hom} _{D}(X,f(-))}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ab90df1b2eca31aca3d08db9350b2ce011eaaf7f"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:33.188ex; height:2.843ex;"" alt=""\operatorname {Hom} _{C}(u_{X},-)\simeq \operatorname {Hom} _{D}(X,f(-))""/></span>, then taking <i>c</i> to be <i>u</i><sub><i>X</i></sub> one gets a universal morphism by sending the identity morphism. In other words, having a universal morphism is equivalent to the representability of the functor <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle \operatorname {Hom} _{D}(X,f(-))}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>D</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo>,</mo>
        <mi>f</mi>
        <mo stretchy=""false"">(</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo stretchy=""false"">)</mo>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle \operatorname {Hom} _{D}(X,f(-))}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/b960a3eee0e23d4564a693fbdeedb1d3f8e50aa6"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:16.154ex; height:2.843ex;"" alt=""\operatorname {Hom} _{D}(X,f(-))""/></span>.</dd>"
"Waldhausen category (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Waldhausen_category"" title=""Waldhausen category"">Waldhausen category</a> is, roughly, a category with families of cofibrations and weak equivalences.
"
"wellpowered (Wikipedia Glossary of Category Theory)"¬"
A category is wellpowered if for each object there is only a set of pairwise non-isomorphic <a href=""https://en.wikipedia.org/wiki/Subobject"" title=""Subobject"">subobjects</a>."
"Yoneda (Wikipedia Glossary of Category Theory)"¬"
<dd class=""glossary"">1.&#160;&#160;<div class=""quotebox pullquote floatright"" style=""width:33%;&#32;;""><link rel=""mw-deduplicated-inline-style"" href=""mw-data:TemplateStyles:r887775652""/>
<div class=""quotebox-quote left-aligned"" style="""">Yoneda’s Lemma asserts ... in more evocative terms, a mathematical object X is best thought of in the context of a category surrounding it, and is determined by the network of relations it enjoys with all the objects of that category. Moreover, to understand X it might be more germane to deal directly with the functor representing it. This is reminiscent of Wittgenstein’s ’language game’; i.e., that the meaning of a word is—in essence—determined by, in fact is nothing more than, its relations to all the utterances in a language.</div>
<p><cite class=""left-aligned"" style=""""><a href=""https://en.wikipedia.org/wiki/Barry_Mazur"" title=""Barry Mazur"">Barry Mazur</a>,&#32;<a rel=""nofollow"" class=""external text"" href=""http://www.math.harvard.edu/~mazur/papers/Thinking.about.Grothendieck%285%29.pdf"">Thinking about Grothendieck</a></cite>
</p>
</div>The <a href=""https://en.wikipedia.org/wiki/Yoneda_lemma"" title=""Yoneda lemma"">Yoneda lemma</a> says: for each set-valued contravariant functor <i>F</i> on <i>C</i> and an object <i>X</i> in <i>C</i>, there is a natural bijection
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F(X)\simeq \operatorname {Nat} (\operatorname {Hom} _{C}(-,X),F)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <mo stretchy=""false"">(</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
        <mo>&#x2243;<!-- ≃ --></mo>
        <mi>Nat</mi>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>C</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo>,</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mi>F</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F(X)\simeq \operatorname {Nat} (\operatorname {Hom} _{C}(-,X),F)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9a3b115e08bc0877787d2a8b8b068a09be0dd3"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:29.977ex; height:2.843ex;"" alt=""{\displaystyle F(X)\simeq \operatorname {Nat} (\operatorname {Hom} _{C}(-,X),F)}""/></span></dd></dl>
<p>where Nat means the set of natural transformations. In particular, the functor
</p>
<dl><dd><span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle y:C\to \mathbf {Fct} (C^{\text{op}},\mathbf {Set} ),\,X\mapsto \operatorname {Hom} _{C}(-,X)}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>y</mi>
        <mo>:</mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">F</mi>
          <mi mathvariant=""bold"">c</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">(</mo>
        <msup>
          <mi>C</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mtext>op</mtext>
          </mrow>
        </msup>
        <mo>,</mo>
        <mrow class=""MJX-TeXAtom-ORD"">
          <mi mathvariant=""bold"">S</mi>
          <mi mathvariant=""bold"">e</mi>
          <mi mathvariant=""bold"">t</mi>
        </mrow>
        <mo stretchy=""false"">)</mo>
        <mo>,</mo>
        <mspace width=""thinmathspace"" />
        <mi>X</mi>
        <mo stretchy=""false"">&#x21A6;<!-- ↦ --></mo>
        <msub>
          <mi>Hom</mi>
          <mrow class=""MJX-TeXAtom-ORD"">
            <mi>C</mi>
          </mrow>
        </msub>
        <mo>&#x2061;<!-- ⁡ --></mo>
        <mo stretchy=""false"">(</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mo>,</mo>
        <mi>X</mi>
        <mo stretchy=""false"">)</mo>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle y:C\to \mathbf {Fct} (C^{\text{op}},\mathbf {Set} ),\,X\mapsto \operatorname {Hom} _{C}(-,X)}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/28abf5bb4c35aee9919dc9f236e691e90671cec8"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.838ex; width:42.711ex; height:2.843ex;"" alt=""{\displaystyle y:C\to \mathbf {Fct} (C^{\text{op}},\mathbf {Set} ),\,X\mapsto \operatorname {Hom} _{C}(-,X)}""/></span></dd></dl>
is fully faithful and is called the Yoneda embedding.<sup id=""cite_ref-15"" class=""reference""><a href=""https://en.wikipedia.org/wiki/Glossary_of_category_theory#cite_note-15"">&#91;15&#93;</a></sup></dd>
<dd class=""glossary"">2.&#160;&#160;If <span class=""mwe-math-element""><span class=""mwe-math-mathml-inline mwe-math-mathml-a11y"" style=""display: none;""><math xmlns=""http://www.w3.org/1998/Math/MathML""  alttext=""{\displaystyle F:C\to D}"">
  <semantics>
    <mrow class=""MJX-TeXAtom-ORD"">
      <mstyle displaystyle=""true"" scriptlevel=""0"">
        <mi>F</mi>
        <mo>:</mo>
        <mi>C</mi>
        <mo stretchy=""false"">&#x2192;<!-- → --></mo>
        <mi>D</mi>
      </mstyle>
    </mrow>
    <annotation encoding=""application/x-tex"">{\displaystyle F:C\to D}</annotation>
  </semantics>
</math></span><img src=""https://wikimedia.org/api/rest_v1/media/math/render/svg/d082755579fdc0f7d9daca655472d59e8347242d"" class=""mwe-math-fallback-image-inline"" aria-hidden=""true"" style=""vertical-align: -0.338ex; width:10.983ex; height:2.176ex;"" alt=""F:C\to D""/></span> is a functor and <i>y</i> is the Yoneda embedding of <i>C</i>, then the <a href=""https://en.wikipedia.org/wiki/Yoneda_extension"" class=""mw-redirect"" title=""Yoneda extension"">Yoneda extension</a> of <i>F</i> is the left Kan extension of <i>F</i> along <i>y</i>.</dd>"
"zero (Wikipedia Glossary of Category Theory)"¬"
A <a href=""https://en.wikipedia.org/wiki/Zero_object"" class=""mw-redirect"" title=""Zero object"">zero object</a> is an object that is both initial and terminal, such as a <a href=""https://en.wikipedia.org/wiki/Trivial_group"" title=""Trivial group"">trivial group</a> in <b><a href=""https://en.wikipedia.org/wiki/Category_of_groups"" title=""Category of groups"">Grp</a></b>.
"