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Merged
ecavallo merged 4 commits into from
Jul 25, 2025
Merged

Strict monoidal categories are strict categories #1196

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9de7ed4
Strict monoidal categories are strict categories #1194
anshwad10 Feb 22, 2025
0a6dbb9
Fix capitalization
anshwad10 Feb 22, 2025
3eb05e5
Remove constructors
anshwad10 Feb 22, 2025
a5e4b92
Add axioms for the morphisms
anshwad10 Jul 25, 2025
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43 changes: 29 additions & 14 deletions Cubical/Categories/Monoidal/Base.agda
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Expand Up @@ -9,25 +9,26 @@ open import Cubical.Categories.Functor.Base
open import Cubical.Categories.Morphism
open import Cubical.Categories.NaturalTransformation.Base
open import Cubical.Foundations.Prelude
open import Cubical.Algebra.Monoid


module _ {ℓ ℓ' : Level} (C : Category ℓ ℓ') where
open Category C

record TensorStr : Type (ℓ-max ℓ ℓ') where
field
─⊗─ : Functor (C ×C C) C
unit : ob
field
─⊗─ : Functor (C ×C C) C
unit : ob

open Functor
open Functor

-- Useful tensor product notation
_⊗_ : ob → ob → ob
x ⊗ y = ─⊗─ .F-ob (x , y)
-- Useful tensor product notation
_⊗_ : ob → ob → ob
x ⊗ y = ─⊗─ .F-ob (x , y)

_⊗ₕ_ : ∀ {x y z w} → Hom[ x , y ] → Hom[ z , w ]
→ Hom[ x ⊗ z , y ⊗ w ]
f ⊗ₕ g = ─⊗─ .F-hom (f , g)
_⊗ₕ_ : ∀ {x y z w} → Hom[ x , y ] → Hom[ z , w ]
→ Hom[ x ⊗ z , y ⊗ w ]
f ⊗ₕ g = ─⊗─ .F-hom (f , g)


record StrictMonStr : Type (ℓ-max ℓ ℓ') where
Expand All @@ -38,10 +39,24 @@ module _ {ℓ ℓ' : Level} (C : Category ℓ ℓ') where

field
-- Axioms - strict
assoc : ∀ x y z → x ⊗ (y ⊗ z) ≡ (x ⊗ y) ⊗ z
idl : ∀ x → unit ⊗ x ≡ x
idr : ∀ x → x ⊗ unit ≡ x
is-monoid : IsMonoid unit _⊗_

open IsMonoid is-monoid renaming (·Assoc to assoc; ·IdL to idl; ·IdR to idr) public

field
-- Axioms for the morphisms
assocₕ : ∀ {x y z x' y' z'}
→ (f : Hom[ x , x' ]) (g : Hom[ y , y' ]) (h : Hom[ z , z' ])
→ (λ i → Hom[ assoc x y z i , assoc x' y' z' i ])
[ f ⊗ₕ (g ⊗ₕ h) ≡ (f ⊗ₕ g) ⊗ₕ h ]

idlₕ : ∀ {x x'} (f : Hom[ x , x' ])
→ (λ i → Hom[ idl x i , idl x' i ])
[ id ⊗ₕ f ≡ f ]

idrₕ : ∀ {x x'} (f : Hom[ x , x' ])
→ (λ i → Hom[ idr x i , idr x' i ])
[ f ⊗ₕ id ≡ f ]

record MonoidalStr : Type (ℓ-max ℓ ℓ') where
field
Expand Down Expand Up @@ -71,7 +86,7 @@ module _ {ℓ ℓ' : Level} (C : Category ℓ ℓ') where

-- More nice notations
α⟨_,_,_⟩ : (x y z : ob) → Hom[ x ⊗ (y ⊗ z) , (x ⊗ y) ⊗ z ]
α⟨ x , y , z ⟩ = α .trans ⟦ ( x , y , z ) ⟧
α⟨ x , y , z ⟩ = α .trans ⟦ (x , y , z) ⟧

η⟨_⟩ : (x : ob) → Hom[ unit ⊗ x , x ]
η⟨ x ⟩ = η .trans ⟦ x ⟧
Expand Down