Add definition of an adjunction in Bicategory

Author: Boarders

src/Categories/Bicategory/Adjunction.agda

@@ -0,0 +1,58 @@
+{-# OPTIONS --without-K --safe #-}
+
+module Categories.Bicategory.Adjunction where
+
+open import Categories.Bicategory
+import Categories.Bicategory.Extras as Extras
+open import Categories.Category
+open import Level
+
+private
+  variable
+    o ℓ e t : Level
+
+module _ (𝒞 : Bicategory o ℓ e t) where
+  open Bicategory 𝒞
+
+  record Adjunction (A B : Obj) : Set (o ⊔ ℓ ⊔ e ⊔ t) where
+    private
+      module C = Extras 𝒞
+
+    field
+      L : A ⇒₁ B
+      R : B ⇒₁ A
+
+    field
+      unit : id₁ ⇒₂ R ⊚₀ L
+      counit : L ⊚₀ R ⇒₂ id₁
+
+    private
+{-
+      L 1 → L (R L) → (L R) L
+                          ↓           L 1
+                        1 ∘ L      ~   ↓
+                          ↓            L
+                          L
+
+-}
+      l-triangle-l : L ⊚₀ id₁ ⇒₂ L
+      l-triangle-l = C.unitorˡ.from ∘ᵥ (counit ⊚₁ id₂) ∘ᵥ C.associator.to ∘ᵥ (id₂ ⊚₁ unit)
+      
+      l-triangle-r : L ⊚₀ id₁ ⇒₂ L
+      l-triangle-r = C.unitorʳ.from
+{-
+      1 R → (R L) R → R (L R)
+                          ↓           L 1
+                        R ∘ id     ~   ↓
+                          ↓            L
+                          R
+   
+-}
+      r-triangle-l : id₁ ⊚₀ R ⇒₂ R
+      r-triangle-l = C.unitorʳ.from ∘ᵥ (id₂ ⊚₁ counit) ∘ᵥ C.associator.from ∘ᵥ (unit ⊚₁ id₂)
+
+      r-triangle-r : id₁ ⊚₀ R ⇒₂ R
+      r-triangle-r = C.unitorˡ.from
+    field
+      l-triangle : l-triangle-l C.≈ l-triangle-r
+      r-triangle : r-triangle-l C.≈ r-triangle-r