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Define adjoint
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@Taneb
Taneb committed Jan 10, 2024
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103 changes: 101 additions & 2 deletions src/Categories/Adjoint/Instance/Slice.agda
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Expand Up @@ -5,10 +5,18 @@ open import Categories.Category using (Category)
module Categories.Adjoint.Instance.Slice {o ℓ e} (C : Category o ℓ e) where

open import Categories.Adjoint using (_⊣_)
open import Categories.Category.Slice C using (SliceObj; Slice⇒; slicearr)
open import Categories.Functor.Slice C using (Forgetful; Free)
open import Categories.Category.BinaryProducts C
open import Categories.Category.Cartesian using (Cartesian)
open import Categories.Category.CartesianClosed using (CartesianClosed)
open import Categories.Category.Slice C using (SliceObj; sliceobj; Slice⇒; slicearr)
open import Categories.Functor.Slice C using (Forgetful; Free; Coforgetful)
open import Categories.Morphism.Reasoning C
open import Categories.NaturalTransformation using (ntHelper)
open import Categories.Diagram.Pullback C hiding (swap)
open import Categories.Object.Product C
open import Categories.Object.Terminal C

open import Function.Base using (_$_)

open Category C
open HomReasoning
Expand Down Expand Up @@ -44,3 +52,94 @@ module _ {A : Obj} (product : ∀ {X} → Product A X) where
⟨ π₁ , π₂ ⟩ ≈⟨ η ⟩
id ∎
}

module _ {A : Obj} (ccc : CartesianClosed C) (pullback : ∀ {X} {Y} {Z} (h : X ⇒ Z) (i : Y ⇒ Z) → Pullback h i) where

open CartesianClosed ccc
open Cartesian cartesian
open Terminal terminal
open BinaryProducts products

Free⊣Coforgetful : Free {A = A} product ⊣ Coforgetful ccc pullback
Free⊣Coforgetful = record
{ unit = ntHelper record
{ η = λ X → p.universal (sliceobj π₁) (λ-unique₂′ (unit-pb X))
; commute = λ {S} {T} f → p.unique-diagram (sliceobj π₁) !-unique₂ $ begin
p.p₂ (sliceobj π₁) ∘ p.universal (sliceobj π₁) _ ∘ f ≈⟨ pullˡ (p.p₂∘universal≈h₂ (sliceobj π₁)) ⟩
λg swap ∘ f ≈⟨ subst ⟩
λg (swap ∘ first f) ≈⟨ λ-cong swap∘⁂ ⟩
λg (second f ∘ swap) ≈˘⟨ λ-cong (∘-resp-≈ʳ β′) ⟩
λg (second f ∘ eval′ ∘ first (λg swap)) ≈˘⟨ λ-cong (∘-resp-≈ʳ (∘-resp-≈ʳ (⁂-cong₂ (p.p₂∘universal≈h₂ (sliceobj π₁)) Equiv.refl))) ⟩
λg (second f ∘ eval′ ∘ first (p.p₂ (sliceobj π₁) ∘ p.universal (sliceobj π₁) _)) ≈˘⟨ λ-cong (pull-last first∘first) ⟩
λg ((second f ∘ eval′ ∘ first (p.p₂ (sliceobj π₁))) ∘ first (p.universal (sliceobj π₁) _)) ≈˘⟨ subst ⟩
λg (second f ∘ eval′ ∘ first (p.p₂ (sliceobj π₁))) ∘ p.universal (sliceobj π₁) _ ≈˘⟨ pullˡ (p.p₂∘universal≈h₂ (sliceobj π₁)) ⟩
p.p₂ (sliceobj π₁) ∘ p.universal (sliceobj π₁) _ ∘ p.universal (sliceobj π₁) _ ∎
}
; counit = ntHelper record
{ η = λ X → slicearr (counit-△ X)
; commute = λ {S} {T} f → begin
(eval′ ∘ first (p.p₂ T) ∘ swap) ∘ second (p.universal T _) ≈⟨ pull-last swap∘⁂ ⟩
eval′ ∘ first (p.p₂ T) ∘ first (p.universal T _) ∘ swap ≈⟨ refl⟩∘⟨ pullˡ first∘first ⟩
eval′ ∘ first (p.p₂ T ∘ p.universal T _) ∘ swap ≈⟨ refl⟩∘⟨ ⁂-cong₂ (p.p₂∘universal≈h₂ T) Equiv.refl ⟩∘⟨refl ⟩
eval′ ∘ first (λg (h f ∘ eval′ ∘ first (p.p₂ S))) ∘ swap ≈⟨ pullˡ β′ ⟩
(h f ∘ eval′ ∘ first (p.p₂ S)) ∘ swap ≈⟨ assoc²' ⟩
h f ∘ eval′ ∘ first (p.p₂ S) ∘ swap ∎
}
; zig = λ {X} → begin
(eval′ ∘ first (p.p₂ (sliceobj π₁)) ∘ swap) ∘ second (p.universal (sliceobj π₁) _) ≈⟨ assoc²' ⟩
eval′ ∘ first (p.p₂ (sliceobj π₁)) ∘ swap ∘ second (p.universal (sliceobj π₁) _) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ swap∘⁂ ⟩
eval′ ∘ first (p.p₂ (sliceobj π₁)) ∘ first (p.universal (sliceobj π₁) _) ∘ swap ≈⟨ refl⟩∘⟨ pullˡ first∘first ⟩
eval′ ∘ first (p.p₂ (sliceobj π₁) ∘ p.universal (sliceobj π₁) _) ∘ swap ≈⟨ refl⟩∘⟨ ⁂-cong₂ (p.p₂∘universal≈h₂ (sliceobj π₁)) Equiv.refl ⟩∘⟨refl ⟩
eval′ ∘ first (λg swap) ∘ swap ≈⟨ pullˡ β′ ⟩
swap ∘ swap ≈⟨ swap∘swap ⟩
id ∎
; zag = λ {X} → p.unique-diagram X !-unique₂ $ begin
p.p₂ X ∘ p.universal X _ ∘ p.universal (sliceobj π₁) _ ≈⟨ pullˡ (p.p₂∘universal≈h₂ X) ⟩
λg ((eval′ ∘ first (p.p₂ X) ∘ swap) ∘ eval′ ∘ first (p.p₂ (sliceobj π₁))) ∘ p.universal (sliceobj π₁) _ ≈˘⟨ pullˡ (subst ○ λ-cong assoc) ⟩
λg ((eval′ ∘ first (p.p₂ X) ∘ swap) ∘ eval′) ∘ p.p₂ (sliceobj π₁) ∘ p.universal (sliceobj π₁) _ ≈⟨ refl⟩∘⟨ p.p₂∘universal≈h₂ (sliceobj π₁) ⟩
λg ((eval′ ∘ first (p.p₂ X) ∘ swap) ∘ eval′) ∘ λg swap ≈⟨ subst ⟩
λg (((eval′ ∘ first (p.p₂ X) ∘ swap) ∘ eval′) ∘ first (λg swap)) ≈⟨ λ-cong (pullʳ β′) ⟩
λg ((eval′ ∘ first (p.p₂ X) ∘ swap) ∘ swap) ≈⟨ λ-cong (pull-last swap∘swap) ⟩
λg (eval′ ∘ first (p.p₂ X) ∘ id) ≈⟨ λ-cong (∘-resp-≈ʳ identityʳ) ⟩
λg (eval′ ∘ first (p.p₂ X)) ≈⟨ η-exp′ ⟩
p.p₂ X ≈˘⟨ identityʳ ⟩
p.p₂ X ∘ id ∎
}
where
p : (X : SliceObj A) → Pullback {X = ⊤} {Z = A ^ A} {Y = Y X ^ A} (λg π₂) (λg (arr X ∘ eval′))
p X = pullback (λg π₂) (λg (arr X ∘ eval′))
module p X = Pullback (p X)

abstract1
unit-pb : ∀ (X : Obj) → eval′ ∘ first {A = X} {C = A} (λg π₂ ∘ !) ≈ eval′ ∘ first (λg (π₁ ∘ eval′) ∘ λg swap)
unit-pb X = begin
eval′ ∘ first (λg π₂ ∘ !) ≈˘⟨ refl⟩∘⟨ first∘first ⟩
eval′ ∘ first (λg π₂) ∘ first ! ≈⟨ pullˡ β′ ⟩
π₂ ∘ first ! ≈⟨ π₂∘⁂ ○ identityˡ ⟩
π₂ ≈˘⟨ project₁ ⟩
π₁ ∘ swap ≈˘⟨ refl⟩∘⟨ β′ ⟩
π₁ ∘ eval′ ∘ first (λg swap) ≈˘⟨ extendʳ β′ ⟩
eval′ ∘ first (λg (π₁ ∘ eval′)) ∘ first (λg swap) ≈⟨ refl⟩∘⟨ first∘first ⟩
eval′ ∘ first (λg (π₁ ∘ eval′) ∘ λg swap) ∎
-- A good chunk of the above, maybe all if you squint, is duplicated with F₁-lemma
-- eval′ ∘ first (λg π₂ ∘ !) ≈ eval′ ∘ first (λg (f ∘ eval′) ∘ first (λg g)
-- With f : X ⇒ Y and g : Z × Y ⇒ X. Not sure what conditions f and g need to have

-- Would it be better if Free used π₂ rather than π₁?
-- It would mean we could avoid this swap
counit-△ : ∀ X → arr X ∘ eval′ ∘ first (p.p₂ X) ∘ swap ≈ π₁
counit-△ X = begin
arr X ∘ eval′ ∘ first (p.p₂ X) ∘ swap ≈˘⟨ assoc² ⟩
((arr X ∘ eval′) ∘ first (p.p₂ X)) ∘ swap ≈⟨ lemma ⟩∘⟨refl ⟩
(π₂ ∘ first (p.p₁ X)) ∘ swap ≈⟨ (π₂∘⁂ ○ identityˡ) ⟩∘⟨refl ⟩
π₂ ∘ swap ≈⟨ project₂ ⟩
π₁ ∎
where
lemma : (arr X ∘ eval′) ∘ first (p.p₂ X) ≈ π₂ ∘ first (p.p₁ X)
lemma = λ-inj $ begin
λg ((arr X ∘ eval′) ∘ first (p.p₂ X)) ≈˘⟨ subst ⟩
λg (arr X ∘ eval′) ∘ p.p₂ X ≈˘⟨ p.commute X ⟩
λg π₂ ∘ p.p₁ X ≈⟨ subst ⟩
λg (π₂ ∘ first (p.p₁ X)) ∎


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