Use alternate syntax for arrow-composition
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@ -53,6 +53,12 @@ record Category (ℓ ℓ' : Level) : Set (lsuc (ℓ' ⊔ ℓ)) where
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open Category
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_[_,_] : ∀ {ℓ ℓ'} → (ℂ : Category ℓ ℓ') → (A : ℂ .Object) → (B : ℂ .Object) → Set ℓ'
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_[_,_] = Arrow
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_[_∘_] : ∀ {ℓ ℓ'} → (ℂ : Category ℓ ℓ') → {A B C : ℂ .Object} → (g : ℂ [ B , C ]) → (f : ℂ [ A , B ]) → ℂ [ A , C ]
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_[_∘_] = _⊕_
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module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} where
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module _ { A B : ℂ .Object } where
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Isomorphism : (f : ℂ .Arrow A B) → Set ℓ'
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@ -11,28 +11,26 @@ open Category
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module _ {ℓc ℓc' ℓd ℓd'} (ℂ : Category ℓc ℓc') (𝔻 : Category ℓd ℓd') where
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record IsFunctor
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(func* : ℂ .Object → 𝔻 .Object)
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(func→ : {A B : ℂ .Object} → ℂ .Arrow A B → 𝔻 .Arrow (func* A) (func* B))
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: Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
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(func→ : {A B : ℂ .Object} → ℂ [ A , B ] → 𝔻 [ func* A , func* B ])
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: Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
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field
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ident : { c : ℂ .Object } → func→ (ℂ .𝟙 {c}) ≡ 𝔻 .𝟙 {func* c}
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-- TODO: Avoid use of ugly explicit arguments somehow.
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-- This guy managed to do it:
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-- https://github.com/copumpkin/categories/blob/master/Categories/Functor/Core.agda
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distrib : {A B C : ℂ .Object} {f : ℂ .Arrow A B} {g : ℂ .Arrow B C}
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→ func→ (ℂ ._⊕_ g f) ≡ 𝔻 ._⊕_ (func→ g) (func→ f)
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distrib : {A B C : ℂ .Object} {f : ℂ [ A , B ]} {g : ℂ [ B , C ]}
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→ func→ (ℂ [ g ∘ f ]) ≡ 𝔻 [ func→ g ∘ func→ f ]
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record Functor : Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
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field
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func* : ℂ .Object → 𝔻 .Object
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func→ : {A B : ℂ .Object} → ℂ .Arrow A B → 𝔻 .Arrow (func* A) (func* B)
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func→ : ∀ {A B} → ℂ [ A , B ] → 𝔻 [ func* A , func* B ]
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{{isFunctor}} : IsFunctor func* func→
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open IsFunctor
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open Functor
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module _ {ℓ ℓ' : Level} {ℂ 𝔻 : Category ℓ ℓ'} where
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private
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_ℂ⊕_ = ℂ ._⊕_
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-- IsFunctor≡ : ∀ {A B : ℂ .Object} {func* : ℂ .Object → 𝔻 .Object} {func→ : {A B : ℂ .Object} → ℂ .Arrow A B → 𝔻 .Arrow (func* A) (func* B)} {F G : IsFunctor ℂ 𝔻 func* func→}
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-- → (eqI : PathP (λ i → ∀ {A : ℂ .Object} → func→ (ℂ .𝟙 {A}) ≡ 𝔻 .𝟙 {func* A})
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@ -45,13 +43,13 @@ module _ {ℓ ℓ' : Level} {ℂ 𝔻 : Category ℓ ℓ'} where
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Functor≡ : {F G : Functor ℂ 𝔻}
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→ (eq* : F .func* ≡ G .func*)
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→ (eq→ : PathP (λ i → ∀ {x y} → ℂ .Arrow x y → 𝔻 .Arrow (eq* i x) (eq* i y))
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→ (eq→ : PathP (λ i → ∀ {x y} → ℂ [ x , y ] → 𝔻 [ eq* i x , eq* i y ])
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(F .func→) (G .func→))
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-- → (eqIsF : PathP (λ i → IsFunctor ℂ 𝔻 (eq* i) (eq→ i)) (F .isFunctor) (G .isFunctor))
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→ (eqI : PathP (λ i → ∀ {A : ℂ .Object} → eq→ i (ℂ .𝟙 {A}) ≡ 𝔻 .𝟙 {eq* i A})
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(F .isFunctor .ident) (G .isFunctor .ident))
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→ (eqD : PathP (λ i → {A B C : ℂ .Object} {f : ℂ .Arrow A B} {g : ℂ .Arrow B C}
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→ eq→ i (ℂ ._⊕_ g f) ≡ 𝔻 ._⊕_ (eq→ i g) (eq→ i f))
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→ (eqD : PathP (λ i → {A B C : ℂ .Object} {f : ℂ [ A , B ]} {g : ℂ [ B , C ]}
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→ eq→ i (ℂ [ g ∘ f ]) ≡ 𝔻 [ eq→ i g ∘ eq→ i f ])
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(F .isFunctor .distrib) (G .isFunctor .distrib))
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→ F ≡ G
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Functor≡ eq* eq→ eqI eqD i = record { func* = eq* i ; func→ = eq→ i ; isFunctor = record { ident = eqI i ; distrib = eqD i } }
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@ -62,17 +60,14 @@ module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : F
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F→ = F .func→
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G* = G .func*
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G→ = G .func→
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_A⊕_ = A ._⊕_
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_B⊕_ = B ._⊕_
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_C⊕_ = C ._⊕_
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module _ {a0 a1 a2 : A .Object} {α0 : A .Arrow a0 a1} {α1 : A .Arrow a1 a2} where
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module _ {a0 a1 a2 : A .Object} {α0 : A [ a0 , a1 ]} {α1 : A [ a1 , a2 ]} where
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dist : (F→ ∘ G→) (α1 A⊕ α0) ≡ (F→ ∘ G→) α1 C⊕ (F→ ∘ G→) α0
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dist : (F→ ∘ G→) (A [ α1 ∘ α0 ]) ≡ C [ (F→ ∘ G→) α1 ∘ (F→ ∘ G→) α0 ]
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dist = begin
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(F→ ∘ G→) (α1 A⊕ α0) ≡⟨ refl ⟩
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F→ (G→ (α1 A⊕ α0)) ≡⟨ cong F→ (G .isFunctor .distrib)⟩
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F→ ((G→ α1) B⊕ (G→ α0)) ≡⟨ F .isFunctor .distrib ⟩
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(F→ ∘ G→) α1 C⊕ (F→ ∘ G→) α0 ∎
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(F→ ∘ G→) (A [ α1 ∘ α0 ]) ≡⟨ refl ⟩
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F→ (G→ (A [ α1 ∘ α0 ])) ≡⟨ cong F→ (G .isFunctor .distrib)⟩
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F→ (B [ G→ α1 ∘ G→ α0 ]) ≡⟨ F .isFunctor .distrib ⟩
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C [ (F→ ∘ G→) α1 ∘ (F→ ∘ G→) α0 ] ∎
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_∘f_ : Functor A C
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_∘f_ =
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