Update category of cospans

1 files changed, 48 insertions, 48 deletions

diff --git a/Functor/Instance/Cospan/Stack.agda b/Functor/Instance/Cospan/Stack.agda
index 03cca1f..b72219b 100644
--- a/Functor/Instance/Cospan/Stack.agda
+++ b/Functor/Instance/Cospan/Stack.agda
@@ -9,9 +9,11 @@ import Categories.Diagram.Pushout.Properties as PushoutProperties
 import Categories.Morphism as Morphism
 import Categories.Morphism.Reasoning as ⇒-Reasoning
 
-open import Categories.Category.Core using (Category)
+open import Categories.Category using (Category)
+open import Categories.Functor using (Functor)
 open import Categories.Functor.Bifunctor using (Bifunctor)
-open import Category.Instance.Cospans 𝒞 using (Cospan; Cospans; Same; id-Cospan; compose)
+open import Category.Instance.Cospans 𝒞 using (Cospans)
+open import Category.Diagram.Cospan 𝒞 as Cospan using (Cospan; identity; compose; _⊗_)
 open import Category.Instance.FinitelyCocompletes {o} {ℓ} {e} using () renaming (_×_ to _×′_)
 open import Category.Cartesian.Instance.FinitelyCocompletes {o} {ℓ} {e} using (-+-; FinitelyCocompletes-CC)
 open import Data.Product.Base using (Σ; _,_; _×_; proj₁; proj₂)
@@ -32,27 +34,19 @@ open DiagramPushout U×U using () renaming (Pushout to Pushout′)
 
 open import Categories.Category.Monoidal.Utilities monoidal using (_⊗ᵢ_)
 
-together :  {A A′ B B′ : Obj} → Cospan A B → Cospan A′ B′ → Cospan (A + A′) (B + B′)
-together A⇒B A⇒B′ = record
-    { f₁ = f₁ A⇒B +₁ f₁ A⇒B′
-    ; f₂ = f₂ A⇒B +₁ f₂ A⇒B′
-    }
-  where
-    open Cospan
-
-id⊗id≈id : {A B : Obj} → Same (together (id-Cospan {A}) (id-Cospan {B})) (id-Cospan {A + B})
+id⊗id≈id : {A B : Obj} → identity {A} ⊗ identity {B} Cospan.≈ identity {A + B}
 id⊗id≈id {A} {B} = record
     { ≅N = ≅.refl
-    ; from∘f₁≈f₁′ = from∘f≈f′
-    ; from∘f₂≈f₂′ = from∘f≈f′
+    ; from∘f₁≈f₁ = from∘f≈f
+    ; from∘f₂≈f₂ = from∘f≈f
     }
   where
     open Morphism U using (module ≅)
     open HomReasoning
     open 𝒞 using (+-η; []-cong₂)
     open coproduct {A} {B} using (i₁; i₂)
-    from∘f≈f′ : id ∘ [ i₁ ∘ id , i₂ ∘ id ] 𝒞.≈ id
-    from∘f≈f′ = begin
+    from∘f≈f : id ∘ [ i₁ ∘ id , i₂ ∘ id ] 𝒞.≈ id
+    from∘f≈f = begin
         id ∘ [ i₁ ∘ id , i₂ ∘ id ]  ≈⟨ identityˡ ⟩
         [ i₁ ∘ id , i₂ ∘ id ]       ≈⟨ []-cong₂ identityʳ identityʳ ⟩
         [ i₁ , i₂ ]                 ≈⟨ +-η ⟩
@@ -64,14 +58,14 @@ homomorphism
     → (B⇒C : Cospan B C)
     → (A⇒B′ : Cospan A′ B′)
     → (B⇒C′ : Cospan B′ C′)
-    → Same (together (compose A⇒B B⇒C) (compose A⇒B′ B⇒C′)) (compose (together A⇒B A⇒B′) (together B⇒C B⇒C′) )
+    → compose A⇒B B⇒C ⊗ compose A⇒B′ B⇒C′ Cospan.≈ compose (A⇒B ⊗ A⇒B′) (B⇒C ⊗ B⇒C′)
 homomorphism A⇒B B⇒C A⇒B′ B⇒C′ = record
     { ≅N = ≅N
-    ; from∘f₁≈f₁′ = from∘f₁≈f₁′
-    ; from∘f₂≈f₂′ = from∘f₂≈f₂′
+    ; from∘f₁≈f₁ = from∘f₁≈f₁
+    ; from∘f₂≈f₂ = from∘f₂≈f₂
     }
   where
-    open Cospan
+    open Cospan.Cospan
     open Pushout
     open HomReasoning
     open ⇒-Reasoning U
@@ -89,56 +83,62 @@ homomorphism A⇒B B⇒C A⇒B′ B⇒C′ = record
     P₃′ = IsPushout⇒Pushout (-+-.F-resp-pushout P₁×P₂.isPushout)
     ≅N : Q P₃′ ≅ Q P₃
     ≅N = up-to-iso P₃′ P₃
-    from∘f₁≈f₁′ : from ≅N ∘ (f₁ (compose A⇒B B⇒C) +₁ f₁ (compose A⇒B′ B⇒C′)) ≈ f₁ (compose (together A⇒B A⇒B′) (together B⇒C B⇒C′))
-    from∘f₁≈f₁′ = begin
+    from∘f₁≈f₁ : from ≅N ∘ (f₁ (compose A⇒B B⇒C) +₁ f₁ (compose A⇒B′ B⇒C′)) ≈ f₁ (compose (A⇒B ⊗ A⇒B′) (B⇒C ⊗ B⇒C′))
+    from∘f₁≈f₁ = begin
         from ≅N ∘ (f₁ (compose A⇒B B⇒C) +₁ f₁ (compose A⇒B′ B⇒C′))  ≈⟨ Equiv.refl ⟩
         from ≅N ∘ ((i₁ P₁ ∘ f₁ A⇒B) +₁ (i₁ P₂ ∘ f₁ A⇒B′))           ≈⟨ refl⟩∘⟨ +₁∘+₁ ⟨
         from ≅N ∘ (i₁ P₁ +₁ i₁ P₂) ∘ (f₁ A⇒B +₁ f₁ A⇒B′)            ≈⟨ Equiv.refl ⟩
-        from ≅N ∘ i₁ P₃′ ∘ f₁ (together A⇒B A⇒B′)                   ≈⟨ pullˡ (universal∘i₁≈h₁ P₃′) ⟩
-        i₁ P₃ ∘ f₁ (together A⇒B A⇒B′)                              ∎
-    from∘f₂≈f₂′ : from ≅N ∘ (f₂ (compose A⇒B B⇒C) +₁ f₂ (compose A⇒B′ B⇒C′)) ≈ f₂ (compose (together A⇒B A⇒B′) (together B⇒C B⇒C′))
-    from∘f₂≈f₂′ = begin
+        from ≅N ∘ i₁ P₃′ ∘ f₁ (A⇒B ⊗ A⇒B′)                          ≈⟨ pullˡ (universal∘i₁≈h₁ P₃′) ⟩
+        i₁ P₃ ∘ f₁ (A⇒B ⊗ A⇒B′)                                     ∎
+    from∘f₂≈f₂ : from ≅N ∘ (f₂ (compose A⇒B B⇒C) +₁ f₂ (compose A⇒B′ B⇒C′)) ≈ f₂ (compose (A⇒B ⊗ A⇒B′) (B⇒C ⊗ B⇒C′))
+    from∘f₂≈f₂ = begin
         from ≅N ∘ (f₂ (compose A⇒B B⇒C) +₁ f₂ (compose A⇒B′ B⇒C′))  ≈⟨ Equiv.refl ⟩
         from ≅N ∘ ((i₂ P₁ ∘ f₂ B⇒C) +₁ (i₂ P₂ ∘ f₂ B⇒C′))           ≈⟨ refl⟩∘⟨ +₁∘+₁ ⟨
         from ≅N ∘ (i₂ P₁ +₁ i₂ P₂) ∘ (f₂ B⇒C +₁ f₂ B⇒C′)            ≈⟨ Equiv.refl ⟩
-        from ≅N ∘ i₂ P₃′ ∘ f₂ (together B⇒C B⇒C′)                   ≈⟨ pullˡ (universal∘i₂≈h₂ P₃′) ⟩
-        i₂ P₃ ∘ f₂ (together B⇒C B⇒C′)                              ∎
+        from ≅N ∘ i₂ P₃′ ∘ f₂ (B⇒C ⊗ B⇒C′)                          ≈⟨ pullˡ (universal∘i₂≈h₂ P₃′) ⟩
+        i₂ P₃ ∘ f₂ (B⇒C ⊗ B⇒C′)                                     ∎
 
 ⊗-resp-≈
     : {A A′ B B′ : Obj}
       {f f′ : Cospan A B}
       {g g′ : Cospan A′ B′}
-    → Same f f′
-    → Same g g′
-    → Same (together f g) (together f′ g′)
+    → f Cospan.≈ f′
+    → g Cospan.≈ g′
+    → f ⊗ g Cospan.≈ f′ ⊗ g′
 ⊗-resp-≈ {_} {_} {_} {_} {f} {f′} {g} {g′} ≈f ≈g = record
     { ≅N = ≈f.≅N ⊗ᵢ ≈g.≅N
-    ; from∘f₁≈f₁′ = from∘f₁≈f₁′
-    ; from∘f₂≈f₂′ = from∘f₂≈f₂′
+    ; from∘f₁≈f₁ = from∘f₁≈f₁
+    ; from∘f₂≈f₂ = from∘f₂≈f₂
     }
   where
     open 𝒞 using (-+-)
-    module ≈f = Same ≈f
-    module ≈g = Same ≈g
+    module ≈f = Cospan._≈_ ≈f
+    module ≈g = Cospan._≈_ ≈g
     open HomReasoning
-    open Cospan
+    open Cospan.Cospan
     open 𝒞 using (+₁-cong₂; +₁∘+₁)
-    from∘f₁≈f₁′ : (≈f.from +₁ ≈g.from) ∘ (f₁ f +₁ f₁ g) ≈ f₁ f′ +₁ f₁ g′
-    from∘f₁≈f₁′ = begin 
+    from∘f₁≈f₁ : (≈f.from +₁ ≈g.from) ∘ (f₁ f +₁ f₁ g) ≈ f₁ f′ +₁ f₁ g′
+    from∘f₁≈f₁ = begin 
         (≈f.from +₁ ≈g.from) ∘ (f₁ f +₁ f₁ g) ≈⟨ +₁∘+₁ ⟩
-        (≈f.from ∘ f₁ f) +₁ (≈g.from ∘ f₁ g)  ≈⟨ +₁-cong₂ (≈f.from∘f₁≈f₁′) (≈g.from∘f₁≈f₁′) ⟩
+        (≈f.from ∘ f₁ f) +₁ (≈g.from ∘ f₁ g)  ≈⟨ +₁-cong₂ ≈f.from∘f₁≈f₁ ≈g.from∘f₁≈f₁ ⟩
         f₁ f′ +₁ f₁ g′                        ∎
-    from∘f₂≈f₂′ : (≈f.from +₁ ≈g.from) ∘ (f₂ f +₁ f₂ g) ≈ f₂ f′ +₁ f₂ g′
-    from∘f₂≈f₂′ = begin 
+    from∘f₂≈f₂ : (≈f.from +₁ ≈g.from) ∘ (f₂ f +₁ f₂ g) ≈ f₂ f′ +₁ f₂ g′
+    from∘f₂≈f₂ = begin 
         (≈f.from +₁ ≈g.from) ∘ (f₂ f +₁ f₂ g) ≈⟨ +₁∘+₁ ⟩
-        (≈f.from ∘ f₂ f) +₁ (≈g.from ∘ f₂ g)  ≈⟨ +₁-cong₂ (≈f.from∘f₂≈f₂′) (≈g.from∘f₂≈f₂′) ⟩
+        (≈f.from ∘ f₂ f) +₁ (≈g.from ∘ f₂ g)  ≈⟨ +₁-cong₂ ≈f.from∘f₂≈f₂ ≈g.from∘f₂≈f₂ ⟩
         f₂ f′ +₁ f₂ g′                        ∎
 
+private
+  ⊗′ : Bifunctor Cospans Cospans Cospans
+  ⊗′ = record
+      { F₀ = λ (A , A′) → A + A′
+      ; F₁ = λ (f , g) → f ⊗ g
+      ; identity = λ { {x , y} → id⊗id≈id {x} {y} }
+      ; homomorphism = λ { {_} {_} {_} {A⇒B , A⇒B′} {B⇒C , B⇒C′} → homomorphism A⇒B B⇒C A⇒B′ B⇒C′ }
+      ; F-resp-≈ = λ (≈f , ≈g) → ⊗-resp-≈ ≈f ≈g
+      }
+
 ⊗ : Bifunctor Cospans Cospans Cospans
-⊗ = record
-    { F₀ = λ { (A , A′) → A + A′ }
-    ; F₁ = λ { (f , g) → together f g }
-    ; identity = λ { {x , y} → id⊗id≈id {x} {y} }
-    ; homomorphism = λ { {_} {_} {_} {A⇒B , A⇒B′} {B⇒C , B⇒C′} → homomorphism A⇒B B⇒C A⇒B′ B⇒C′ }
-    ; F-resp-≈ = λ { (≈f , ≈g) → ⊗-resp-≈ ≈f ≈g }
-    }
+⊗ = ⊗′
+
+module ⊗ = Functor ⊗