End calculus

src/Categories/Category/Construction/Functors.agda

@@ -130,6 +130,8 @@ module curry {o₁ e₁ ℓ₁} {C₁ : Category o₁ e₁ ℓ₁}
   open Category
   open NaturalIsomorphism
 
+  module ₀ (F : Bifunctor C₁ C₂ D) = Functor (Functor.F₀ curry F)
+
   -- Currying preserves natural isos.
   -- This makes |curry.F₀| a map between the hom-setoids of Cats.
 
@@ -186,6 +188,12 @@ uncurry {C₁ = C₁} {C₂ = C₂} {D = D} = record
           } where module t = NaturalTransformation t
                   open NaturalTransformation
 
+module uncurry {o₁ e₁ ℓ₁} {C₁ : Category o₁ e₁ ℓ₁}
+               {o₂ e₂ ℓ₂} {C₂ : Category o₂ e₂ ℓ₂}
+               {o′ e′ ℓ′} {D  : Category o′ e′ ℓ′}
+               where
+  open Functor (uncurry {C₁ = C₁} {C₂ = C₂} {D = D}) public
+
 module _ {o₁ e₁ ℓ₁} {C₁ : Category o₁ e₁ ℓ₁}
          {o₂ e₂ ℓ₂} {C₂ : Category o₂ e₂ ℓ₂}
          {o′ e′ ℓ′} {D  : Category o′ e′ ℓ′} where

src/Categories/Category/Construction/TwistedArrow.agda

@@ -1,4 +1,5 @@
 {-# OPTIONS --without-K --safe #-}
+
 open import Categories.Category using (Category; module Definitions)
 
 -- Definition of the "Twisted Arrow" Category of a Category 𝒞
@@ -9,14 +10,16 @@ open import Data.Product using (_,_; _×_; map; zip)
 open import Function.Base using (_$_; flip)
 open import Relation.Binary.Core using (Rel)
 
-import Categories.Morphism as M
-open M 𝒞
-open import Categories.Morphism.Reasoning 𝒞
+open import Categories.Category.Product renaming (Product to _×ᶜ_)
+open import Categories.Functor using (Functor)
+open import Categories.Morphism.Reasoning 𝒞 using (pullˡ; pullʳ )
 
-open Category 𝒞
+private
+  open module 𝒞 = Category 𝒞
 open Definitions 𝒞
 open HomReasoning
 
+
 private
   variable
     A B C : Obj
@@ -68,3 +71,16 @@ TwistedArrow = record
     cod⇒ m₁ ∘ (cod⇒ m₂ ∘ Morphism.arr A) ∘ (dom⇐ m₂ ∘ dom⇐ m₁) ≈⟨ refl⟩∘⟨ (pullˡ assoc) ⟩
     cod⇒ m₁ ∘ (cod⇒ m₂ ∘ Morphism.arr A ∘ dom⇐ m₂) ∘ dom⇐ m₁   ≈⟨ (refl⟩∘⟨ square m₂ ⟩∘⟨refl) ⟩
     cod⇒ m₁ ∘ Morphism.arr B ∘ dom⇐ m₁ ∎
+
+
+-- Consider TwistedArrow as the comma category * / Hom[C][-,-]
+-- We have the codomain functor TwistedArrow → C.op × C
+open Functor
+Codomain : Functor TwistedArrow (𝒞.op ×ᶜ 𝒞)
+Codomain .F₀ x = dom , cod
+  where open Morphism x
+Codomain .F₁ f = dom⇐ , cod⇒
+  where open Morphism⇒ f
+Codomain .identity = Equiv.refl , Equiv.refl
+Codomain .homomorphism = Equiv.refl , Equiv.refl
+Codomain .F-resp-≈ e = e

src/Categories/Diagram/End/Fubini.agda

@@ -0,0 +1,222 @@
+{-# OPTIONS --without-K --lossy-unification --safe #-}
+
+module Categories.Diagram.End.Fubini where
+
+open import Level
+open import Data.Product using (Σ; _,_; _×_) renaming (proj₁ to fst; proj₂ to snd)
+open import Function using (_$_)
+
+open import Categories.Category using (Category; _[_,_]; _[_∘_])
+open import Categories.Category.Construction.Functors using (Functors; curry)
+open import Categories.Category.Product using (πˡ; πʳ; _⁂_; _※_; Swap) renaming (Product to _×ᶜ_)
+open import Categories.Diagram.End using () renaming (End to ∫)
+open import Categories.Diagram.End.Properties using (end-η-commute; end-unique)
+open import Categories.Diagram.End.Parameterized using () renaming (EndF to ⨏)
+open import Categories.Diagram.Wedge using (Wedge)
+open import Categories.Functor using (Functor; _∘F_) renaming (id to idF)
+open import Categories.Functor.Bifunctor using (Bifunctor; appˡ)
+open import Categories.NaturalTransformation using (NaturalTransformation)
+open import Categories.NaturalTransformation.Dinatural using (DinaturalTransformation; dtHelper)
+
+import Categories.Morphism as M
+import Categories.Morphism.Reasoning as MR
+
+variable
+  o ℓ e : Level
+  C P D : Category o ℓ e
+
+module Functor-Swaps  (F : Bifunctor (Category.op C ×ᶜ Category.op P) (C ×ᶜ P) D) where
+  private
+    module C = Category C
+    module P = Category P
+
+  -- just shuffling arguments
+  -- An end of F ′ can be taken iterated, instead of in the product category
+  _′ : Bifunctor (P.op ×ᶜ P) (C.op ×ᶜ C) D
+  _′ = F ∘F ((πˡ ∘F πʳ ※ πˡ ∘F πˡ) ※ (πʳ ∘F πʳ ※ πʳ ∘F πˡ))
+
+  -- An end of F ′′ is the same as F, but the order in the product category is reversed
+  _′′ : Bifunctor (Category.op P ×ᶜ Category.op C) (P ×ᶜ C) D
+  _′′ = F ∘F (Swap ⁂ Swap)
+
+open Functor-Swaps
+
+module _  (F : Bifunctor (Category.op C ×ᶜ Category.op P) (C ×ᶜ P) D)
+          {ω : ∀ ((p , q) : Category.Obj P × Category.Obj P) → ∫ (appˡ (F ′) (p , q))}
+           where
+  open M D
+  private
+    module C = Category C
+    module P = Category P
+    module D = Category D
+    C×P = (C ×ᶜ P)
+    module C×P = Category C×P
+    open module F = Functor F using (F₀; F₁; F-resp-≈)
+    ∫F = (⨏ (F ′) {ω})
+    module ∫F = Functor ∫F
+
+    module ω ((p , q) : P.Obj × P.Obj) = ∫ (ω (p , q))
+
+  open _≅_
+  open Iso
+  open DinaturalTransformation
+  open NaturalTransformation using (η)
+  open Wedge renaming (E to apex)
+  open Category D
+  open import Categories.Morphism.Reasoning D
+  open HomReasoning
+  -- we show that wedges of (∫.E e₀) are in bijection with wedges of F
+  -- following MacLane §IX.8
+  private module _ (ρ : Wedge ∫F) where
+    private
+      open module ρ = Wedge ρ using () renaming (E to x)
+    -- first, our wedge gives us a wedge in the product
+    ξ : Wedge F
+    ξ .apex = x
+    ξ .dinatural .α (c , p) = ω.dinatural.α (p , p) c ∘ ρ.dinatural.α p
+    -- unfolded because dtHelper means opening up a record (i.e. no where clause)
+    ξ .dinatural .op-commute (f , s) = assoc ○ ⟺ (ξ .dinatural .commute (f , s)) ○ sym-assoc
+    ξ .dinatural .commute {c , p} {d , q} (f , s) = begin
+          F₁ (C×P.id , (f , s)) ∘ (ωp,p c ∘ ρp) ∘ id
+            ≈⟨ refl⟩∘⟨ identityʳ ⟩
+          -- First we show dinaturality in C, which is 'trivial'
+          F₁ (C×P.id , (f , s)) ∘ (ωp,p c ∘ ρp)
+            ≈⟨ F-resp-≈ ((C.Equiv.sym C.identity² , P.Equiv.sym P.identity²) , (C.Equiv.sym C.identityˡ , P.Equiv.sym P.identityʳ)) ⟩∘⟨refl ⟩
+          F₁ (C×P [ C×P.id ∘ C×P.id ] , C [ C.id ∘ f ] , P [ s ∘ P.id ]) ∘ (ωp,p c ∘ ρ.dinatural.α p)
+            ≈⟨ F.homomorphism ⟩∘⟨refl ⟩
+          (F₁ (C×P.id , (C.id , s)) ∘ F₁ (C×P.id , (f , P.id))) ∘ (ωp,p c ∘ ρp)
+            ≈⟨ assoc²γβ ⟩
+          (F₁ (C×P.id , (C.id , s)) ∘ F₁ (C×P.id , (f , P.id)) ∘ ωp,p c) ∘ ρp
+            ≈⟨ (refl⟩∘⟨ refl⟩∘⟨ ⟺ identityʳ) ⟩∘⟨refl ⟩
+          (F₁ (C×P.id , (C.id , s)) ∘ F₁ (C×P.id , (f , P.id)) ∘ ωp,p c ∘ id) ∘ ρp
+            ≈⟨ (refl⟩∘⟨ ω.dinatural.commute (p , p) f) ⟩∘⟨refl ⟩
+          (F₁ (C×P.id , (C.id , s)) ∘ F₁ ((f , P.id) , C×P.id) ∘ ωp,p d ∘ id) ∘ ρp
+            ≈⟨ extendʳ (⟺ F.homomorphism ○ F-resp-≈ ((MR.id-comm C , P.Equiv.refl) , (C.Equiv.refl , MR.id-comm P)) ○ F.homomorphism) ⟩∘⟨refl ⟩
+          -- now, we must show dinaturality in P
+          -- First, we get rid of the identity and reassoc
+          (F₁ ((f , P.id) , C×P.id) ∘ F₁ (C×P.id , (C.id , s)) ∘ ωp,p d ∘ id) ∘ ρp
+            ≈⟨ assoc²βε ⟩
+          F₁ ((f , P.id) , C×P.id) ∘ F₁ (C×P.id , (C.id , s)) ∘ (ωp,p d ∘ id)  ∘ ρp
+            ≈⟨ refl⟩∘⟨ refl⟩∘⟨ identityʳ ⟩∘⟨refl ⟩
+          F₁ ((f , P.id) , C×P.id) ∘ F₁ (C×P.id , (C.id , s)) ∘ ωp,p d ∘ ρp
+          -- this is the hard part: we use commutativity from the bottom face of the cube.
+          -- The fact that this exists (an arrow between ∫ F (p,p,- ,-) to ∫ F (p,q,- ,-)) is due to functorality of ∫ and curry₀.
+          -- The square commutes by universiality of ω
+            ≈⟨ refl⟩∘⟨ extendʳ (⟺ (end-η-commute ⦃ ω (p , p) ⦄ ⦃ ω (p , q) ⦄ (curry.₀.₁ (F ′) (P.id , s)) d)) ⟩
+          -- now we can use dinaturality in ρ, which annoyingly has an `id` tacked on the end
+          F₁ ((f , P.id) , C×P.id) ∘ ωp,q d ∘ ∫F.F₁ (P.id , s) ∘ ρp
+            ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨  ⟺ identityʳ ⟩
+          F₁ ((f , P.id) , C×P.id) ∘ ωp,q d ∘ ∫F.F₁ (P.id , s) ∘ ρp ∘ id
+            ≈⟨ refl⟩∘⟨ refl⟩∘⟨ ρ.dinatural.commute s ⟩
+          F₁ ((f , P.id) , C×P.id) ∘ ωp,q d ∘ ∫F.F₁ (s , P.id) ∘ ρq ∘ id
+            ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ identityʳ ⟩
+          F₁ ((f , P.id) , C×P.id) ∘ ωp,q d ∘ ∫F.F₁ (s , P.id) ∘ ρq
+          -- and now we're done. step backward through the previous isomorphisms to get dinaturality in f and s together
+            ≈⟨ refl⟩∘⟨ extendʳ (end-η-commute ⦃ ω (q , q) ⦄ ⦃ ω (p , q) ⦄ (curry.₀.₁ (F ′) (s , P.id)) d) ⟩
+          F₁ ((f , P.id) , C×P.id) ∘ (F₁ ((C.id , s) , C×P.id) ∘ ωq,q d ∘ ρq)
+            ≈⟨ pullˡ (⟺ F.homomorphism ○ F-resp-≈ ((C.identityˡ , P.identityʳ) , (C.identity² , P.identity²))) ⟩
+          F₁ ((f , s) , C×P.id) ∘ ωq,q d ∘ ρq
+            ≈˘⟨ refl⟩∘⟨ identityʳ ⟩
+          F₁ ((f , s) , C×P.id) ∘ (ωq,q d ∘ ρq) ∘ id
+          ∎
+      where ωp,p = ω.dinatural.α (p , p)
+            ωp,q = ω.dinatural.α (p , q)
+            ωq,q = ω.dinatural.α (q , q)
+            ρp = ρ.dinatural.α p
+            ρq = ρ.dinatural.α q
+
+
+  -- now the opposite direction
+  private module _ (ξ : Wedge F) where
+    private
+      open module ξ = Wedge ξ using () renaming (E to x)
+    ρ : Wedge ∫F
+    ρ .apex = x
+    ρ .dinatural .α p = ω.factor (p , p) record
+        { E = x
+        ; dinatural = dtHelper record
+          { α = λ c → ξ.dinatural.α (c , p)
+          ; commute = λ f → ξ.dinatural.commute (f , P.id)
+          }
+        }
+    ρ .dinatural .op-commute f = assoc ○ ⟺ (ρ .dinatural .commute f) ○ sym-assoc
+    ρ .dinatural .commute {p} {q} f = ω.unique′ (p , q) λ {c} → begin
+        ωp,q c ∘ ∫F.₁ (P.id , f) ∘ ρp ∘ id
+          ≈⟨ refl⟩∘⟨ refl⟩∘⟨ identityʳ ⟩
+        ωp,q c ∘ ∫F.₁ (P.id , f) ∘ ρp
+          ≈⟨ extendʳ $ end-η-commute ⦃ ω (p , p) ⦄ ⦃ ω (p , q) ⦄ (curry.₀.₁ (F ′) (P.id , f)) c ⟩
+        F₁ (C×P.id , (C.id , f)) ∘ ωp,p c ∘ ρp
+          ≈⟨ refl⟩∘⟨ ω.universal (p , p) ⟩
+        F₁ (C×P.id , (C.id , f)) ∘ ξ.dinatural.α (c , p)
+          ≈˘⟨ refl⟩∘⟨ identityʳ ⟩
+        F₁ (C×P.id , (C.id , f)) ∘ ξ.dinatural.α (c , p) ∘ id
+          ≈⟨ ξ.dinatural.commute (C.id , f) ⟩
+        F₁ ((C.id , f) , C×P.id) ∘ ξ.dinatural.α (c , q) ∘ id
+          ≈⟨ refl⟩∘⟨ identityʳ ⟩
+        F₁ ((C.id , f) , C×P.id) ∘ ξ.dinatural.α (c , q)
+          ≈˘⟨ refl⟩∘⟨ ω.universal (q , q) ⟩
+        F₁ ((C.id , f) , C×P.id) ∘ ωq,q c ∘ ρq
+          ≈˘⟨ extendʳ $ end-η-commute ⦃ ω (q , q) ⦄ ⦃ ω (p , q) ⦄ (curry.₀.₁ (F ′) (f , P.id)) c ⟩
+        ωp,q c ∘ ∫F.₁ (f , P.id) ∘ ρq
+          ≈˘⟨ refl⟩∘⟨ refl⟩∘⟨ identityʳ ⟩
+        ωp,q c ∘ ∫F.₁ (f , P.id) ∘ ρq ∘ id
+          ∎
+      where ωp,p = ω.dinatural.α (p , p)
+            ωp,q = ω.dinatural.α (p , q)
+            ωq,q = ω.dinatural.α (q , q)
+            ρp = ρ  .dinatural.α p
+            ρq = ρ  .dinatural.α q
+
+  -- This construction is actually stronger than Fubini. It states that the
+  -- iterated end _is_ an end in the product category.
+  -- Thus the product end need not exist.
+  module _ {{eᵢ : ∫ (⨏ (F ′) {ω})}} where
+    private
+      module eᵢ = ∫ eᵢ
+    eₚ : ∫ F
+    eₚ .∫.wedge = ξ eᵢ.wedge
+    eₚ .∫.factor W = eᵢ.factor (ρ W)
+    eₚ .∫.universal {W} {c , p} = begin
+      ξ eᵢ.wedge .dinatural.α (c , p) ∘ eᵢ.factor (ρ W)               ≡⟨⟩
+      (ω.dinatural.α (p , p) c ∘ eᵢ.dinatural.α p)  ∘ eᵢ.factor (ρ W) ≈⟨ pullʳ (eᵢ.universal {ρ W} {p}) ⟩
+      ω.dinatural.α (p , p) c ∘ (ρ W) .dinatural.α p                  ≈⟨ ω.universal (p , p) ⟩
+      W .dinatural.α  (c , p)                                         ∎
+    eₚ .∫.unique {W} {g} h = eᵢ.unique λ {A = p} → Equiv.sym $ ω.unique (p , p) (sym-assoc ○ h)
+    module eₚ = ∫ eₚ
+
+    -- corollary: if a product end exists, it is iso
+    Fubini : {eₚ' : ∫ F} → ∫.E eᵢ ≅ ∫.E eₚ'
+    Fubini {eₚ'} = end-unique eₚ eₚ'
+
+  -- TODO show that an end ∫ F yields an end of ∫ (⨏ (F ′) {ω})
+
+module _ (F : Bifunctor (Category.op C ×ᶜ Category.op P) (C ×ᶜ P) D)
+         {ωᴾ : ∀ ((p , q) : Category.Obj P × Category.Obj P) → ∫ (appˡ (F ′) (p , q))}
+         {ωᶜ : ∀ ((c , d) : Category.Obj C × Category.Obj C) → ∫ (appˡ (F ′′ ′)(c , d))}
+         {{e₁ : ∫ (⨏ (F ′) {ωᴾ})}} {{e₂ : ∫ (⨏ (F ′′ ′) {ωᶜ})}} where
+  open M D
+  -- since `F` and `F′′` are functors from different categories we can't construct a
+  -- natural isomorphism between them. Instead it is best to show that eₚ F is an end in
+  -- ∫ (F ′′) or vice-versa, and then Fubini follows due to uniqueness of ends
+  open Wedge renaming (E to apex)
+  private
+    eₚ′′ : ∫ (F ′′)
+    eₚ′′  .∫.wedge .apex = eₚ.wedge.E F
+    eₚ′′  .∫.wedge .dinatural = dtHelper record
+      { α = λ (p , c) → eₚ.dinatural.α F (c , p)
+      ; commute = λ (s , f) → eₚ.dinatural.commute F (f , s)
+      }
+    eₚ′′  .∫.factor W = eₚ.factor F record
+      { W ; dinatural = dtHelper record
+        { α = λ (c , p) → W.dinatural.α (p , c)
+        ; commute = λ (f , s) → W.dinatural.commute (s , f)
+        }
+      }
+      where module W = Wedge W
+    eₚ′′  .∫.universal {W} {p , c} = eₚ.universal F
+    eₚ′′  .∫.unique {W} {g} h = eₚ.unique F h
+
+  Fubini′ : ∫.E e₁ ≅ ∫.E e₂
+  Fubini′ = ≅.trans (Fubini F {ωᴾ} {eₚ' = eₚ F}) $
+            ≅.trans (end-unique eₚ′′ (eₚ (F ′′)))
+                    (≅.sym (Fubini (F ′′) {ωᶜ} {eₚ' = eₚ (F ′′)}))

src/Categories/Diagram/End/Instances/NaturalTransformation.agda

@@ -0,0 +1,66 @@
+{-# OPTIONS --without-K --safe #-}
+
+open import Data.Product using (Σ; _,_)
+open import Function using (_$_)
+open import Level
+
+open import Categories.Category
+open import Categories.Category.Instance.Setoids
+open import Categories.Category.Product
+open import Categories.Diagram.End using () renaming (End to ∫)
+open import Categories.Functor renaming (id to idF)
+open import Categories.Functor.Hom
+open import Categories.NaturalTransformation renaming (id to idN)
+open import Categories.NaturalTransformation.Dinatural using (DinaturalTransformation; dtHelper)
+open import Categories.NaturalTransformation.Equivalence
+open import Function.Bundles using (Func)
+
+import Categories.Diagram.Wedge as Wedges
+import Categories.Morphism as M
+import Categories.Morphism.Reasoning as MR
+
+module Categories.Diagram.End.Instances.NaturalTransformation {ℓ e o′}
+  {C : Category ℓ ℓ e} {D : Category o′ ℓ ℓ} (F G : Functor C D) where
+
+private
+  module C = Category C
+  module D = Category D
+  open module F = Functor F using (F₀; F₁)
+  module G = Functor G
+
+open Wedges (Hom[ D ][-,-] ∘F (F.op ⁂ G))
+open D
+open HomReasoning
+open MR D
+open NaturalTransformation renaming (η to app)
+open ∫ hiding (dinatural)
+open Wedge
+open Func
+
+NTs-are-End : ∫ (Hom[ D ][-,-] ∘F (F.op ⁂ G))
+NTs-are-End .wedge .E = (≃-setoid F G)
+NTs-are-End .wedge .dinatural = dtHelper record
+  { α = λ C → record
+    { to = λ η → η .app C
+    ; cong = λ e → e
+    }
+  ; commute = λ {X} {Y} f {η} → begin
+    G.₁ f ∘ η .app X ∘ F₁ C.id  ≈⟨ pullˡ (η .sym-commute f) ⟩
+    (η .app Y ∘ F₁ f) ∘ F₁ C.id ≈⟨ refl⟩∘⟨ F.identity ⟩
+    (η .app Y ∘ F₁ f) ∘ id      ≈⟨ id-comm ⟩
+    id ∘ η .app Y ∘ F₁ f        ≈⟨ ⟺ G.identity ⟩∘⟨refl ⟩
+    G.₁ C.id ∘ η .app Y ∘ F₁ f  ∎
+  }
+NTs-are-End .factor W .to s = ntHelper record
+  { η = λ X → W.dinatural.α X .to s
+  -- basically the opposite proof of above
+  ; commute = λ {X} {Y} f → begin
+    W.dinatural.α Y .to s ∘ F₁ f            ≈⟨ introˡ G.identity ⟩
+    G.₁ C.id ∘ W.dinatural.α Y .to s ∘ F₁ f ≈˘⟨ W.dinatural.commute f ⟩
+    G.₁ f ∘ W.dinatural.α X .to s ∘ F₁ C.id ≈⟨ refl⟩∘⟨ elimʳ F.identity ⟩
+    G.F₁ f ∘ W.dinatural.α X .to s          ∎
+  }
+  where module W = Wedge W
+NTs-are-End .factor W .cong e {x} = W .dinatural.α x .cong e
+NTs-are-End .universal = Equiv.refl
+NTs-are-End .unique h = Equiv.sym h