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JacquesCarette merged 1 commit into from
Dec 11, 2025
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Add [_]-resp-inverse #500

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9 changes: 3 additions & 6 deletions src/Categories/Adjoint/Properties.agda
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Expand Up @@ -20,7 +20,7 @@ open import Categories.Functor using (Functor; _∘F_) renaming (id to idF)
open import Categories.Functor.Hom using (Hom[_][-,-])
open import Categories.Functor.Construction.Constant using (const!)
open import Categories.Functor.Construction.LiftSetoids using (LiftSetoids)
open import Categories.Functor.Properties using ([_]-resp-square; [_]-resp-∘)
open import Categories.Functor.Properties using ([_]-resp-square; [_]-resp-inverse)
open import Categories.Functor.Limits using (Continuous; Cocontinuous)
open import Categories.Functor.Bifunctor using (Bifunctor; appʳ; appˡ)
open import Categories.Functor.Bifunctor.Properties using ([_]-decompose₁; [_]-decompose₂; [_]-commute)
Expand Down Expand Up @@ -206,7 +206,7 @@ module _ {L : Functor J K} {R : Functor K J} (L⊣R : L ⊣ R) {F : Functor K C}
; commute = λ {X} → begin
proj (L.₀ X) ∘ rep A′ ≈⟨ commute ⟩
F.₁ (counit.η (L.₀ X)) ∘ A.ψ (R.₀ (L.₀ X)) ≈⟨ refl⟩∘⟨ A.commute (unit.η X) ⟨
F.₁ (counit.η (L.₀ X)) ∘ F.₁ (L.₁ (unit.η X)) ∘ A.ψ X ≈⟨ cancelˡ ([ F ]-resp-∘ zig ○ F.identity) ⟩
F.₁ (counit.η (L.₀ X)) ∘ F.₁ (L.₁ (unit.η X)) ∘ A.ψ X ≈⟨ cancelˡ ([ F ]-resp-inverse zig) ⟩
A.ψ X ∎
}

Expand Down Expand Up @@ -266,10 +266,7 @@ module _ {L : Functor C D} {R : Functor D C} (L⊣R : L ⊣ R) where
}
; assoc = [ R ]-resp-square (counit.commute _)
; sym-assoc = [ R ]-resp-square (counit.sym-commute _)
; identityˡ = λ {X} → begin
μ′.η X C.∘ R.F₁ (L.F₁ (unit.η X)) ≈⟨ [ R ]-resp-∘ zig ⟩
R.F₁ D.id ≈⟨ R.identity ⟩
C.id ∎
; identityˡ = [ R ]-resp-inverse zig
; identityʳ = zag
}
where open C.HomReasoning
Expand Down
12 changes: 4 additions & 8 deletions src/Categories/Adjoint/TwoSided.agda
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Expand Up @@ -54,17 +54,15 @@ record _⊣⊢_ (L : Functor C D) (R : Functor D C) : Set (levelOfTerm L ⊔ lev
zag⁻¹ : {B : D.Obj} → unit.⇐.η (R.₀ B) C.∘ R.₁ (counit.⇐.η B) C.≈ C.id
zag⁻¹ {B} = begin
unit.⇐.η (R.₀ B) C.∘ R.₁ (counit.⇐.η B) ≈˘⟨ flip-fromʳ unit.FX≅GX zag ⟩∘⟨refl ⟩
R.₁ (counit.⇒.η B) C.∘ R.₁ (counit.⇐.η B) ≈⟨ [ R ]-resp-∘ (counit.iso.isoʳ B) ⟩
R.₁ D.id ≈⟨ R.identity ⟩
R.₁ (counit.⇒.η B) C.∘ R.₁ (counit.⇐.η B) ≈⟨ [ R ]-resp-inverse (counit.iso.isoʳ B) ⟩
C.id ∎
where open C.HomReasoning
open MR C

zig⁻¹ : {A : C.Obj} → L.₁ (unit.⇐.η A) D.∘ counit.⇐.η (L.₀ A) D.≈ D.id
zig⁻¹ {A} = begin
L.₁ (unit.⇐.η A) D.∘ counit.⇐.η (L.₀ A) ≈˘⟨ refl⟩∘⟨ flip-fromˡ counit.FX≅GX zig ⟩
L.₁ (unit.⇐.η A) D.∘ L.₁ (unit.⇒.η A) ≈⟨ [ L ]-resp-∘ (unit.iso.isoˡ A) ⟩
L.₁ C.id ≈⟨ L.identity ⟩
L.₁ (unit.⇐.η A) D.∘ L.₁ (unit.⇒.η A) ≈⟨ [ L ]-resp-inverse (unit.iso.isoˡ A) ⟩
D.id ∎
where open D.HomReasoning
open MR D
Expand Down Expand Up @@ -124,8 +122,7 @@ private
helper = begin
R.₁ (L.₁ (R.₁ (counit.⇒.η B) ∘ unit.⇒.η (R.₀ B))) ≈⟨ Functor.homomorphism (R ∘F L) ⟩
R.₁ (L.₁ (R.₁ (counit.⇒.η B))) ∘ R.₁ (L.₁ (unit.⇒.η (R.₀ B))) ≈˘⟨ R.F-resp-≈ (F≃id-comm₁ counit) ⟩∘⟨refl ⟩
R.₁ (counit.⇒.η (L.₀ (R.₀ B))) ∘ R.₁ (L.₁ (unit.⇒.η (R.₀ B))) ≈⟨ [ R ]-resp-∘ zig ⟩
R.₁ D.id ≈⟨ R.identity ⟩
R.₁ (counit.⇒.η (L.₀ (R.₀ B))) ∘ R.₁ (L.₁ (unit.⇒.η (R.₀ B))) ≈⟨ [ R ]-resp-inverse zig ⟩
id ∎

record WithZag (L : Functor C D) (R : Functor D C) : Set (levelOfTerm L ⊔ levelOfTerm R) where
Expand All @@ -152,8 +149,7 @@ private
helper = begin
L.₁ (R.₁ (counit.⇒.η (L.₀ A) ∘ L.₁ (unit.⇒.η A))) ≈⟨ Functor.homomorphism (L ∘F R) ⟩
(L.₁ (R.₁ (counit.⇒.η (L.₀ A))) ∘ L.₁ (R.₁ (L.₁ (unit.⇒.η A)))) ≈˘⟨ refl⟩∘⟨ L.F-resp-≈ (F≃id-comm₂ (≃.sym unit)) ⟩
L.₁ (R.₁ (counit.⇒.η (L.₀ A))) ∘ L.₁ (unit.⇒.η (R.₀ (L.₀ A))) ≈⟨ [ L ]-resp-∘ zag ⟩
L.₁ C.id ≈⟨ L.identity ⟩
L.₁ (R.₁ (counit.⇒.η (L.₀ A))) ∘ L.₁ (unit.⇒.η (R.₀ (L.₀ A))) ≈⟨ [ L ]-resp-inverse zag ⟩
id ∎

module _ {L : Functor C D} {R : Functor D C} where
Expand Down
26 changes: 7 additions & 19 deletions src/Categories/Adjoint/TwoSided/Compose.agda
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Expand Up @@ -68,22 +68,16 @@ _∘⊣⊢_ {C = C} {D} {E} {L} {R} {L′} {R′} L⊣⊢R L′⊣⊢R′ = with
; iso = λ c → record
{ isoˡ = begin
(⊣⊢₁.unit.⇐.η c ∘ R.₁ (⊣⊢₂.unit.⇐.η (L.₀ c))) ∘ R.₁ (⊣⊢₂.unit.⇒.η (L.₀ c)) ∘ ⊣⊢₁.unit.⇒.η c
≈⟨ center ([ R ]-resp-∘ (⊣⊢₂.unit.iso.isoˡ (L.₀ c))) ⟩
⊣⊢₁.unit.⇐.η c ∘ R.₁ D.id ∘ ⊣⊢₁.unit.⇒.η c
≈⟨ refl⟩∘⟨ elimˡ R.identity ⟩
≈⟨ cancelInner ([ R ]-resp-inverse (⊣⊢₂.unit.iso.isoˡ (L.₀ c))) ⟩
⊣⊢₁.unit.⇐.η c ∘ ⊣⊢₁.unit.⇒.η c
≈⟨ ⊣⊢₁.unit.iso.isoˡ c ⟩
id
∎
; isoʳ = begin
(R.₁ (⊣⊢₂.unit.⇒.η (L.₀ c)) ∘ ⊣⊢₁.unit.⇒.η c) ∘ ⊣⊢₁.unit.⇐.η c ∘ R.₁ (⊣⊢₂.unit.⇐.η (L.₀ c))
≈⟨ center (⊣⊢₁.unit.iso.isoʳ c) ⟩
R.₁ (⊣⊢₂.unit.⇒.η (L.₀ c)) ∘ id ∘ R.₁ (⊣⊢₂.unit.⇐.η (L.₀ c))
≈⟨ refl⟩∘⟨ identityˡ ⟩
≈⟨ cancelInner (⊣⊢₁.unit.iso.isoʳ c) ⟩
R.₁ (⊣⊢₂.unit.⇒.η (L.₀ c)) ∘ R.₁ (⊣⊢₂.unit.⇐.η (L.₀ c))
≈⟨ [ R ]-resp-∘ (⊣⊢₂.unit.iso.isoʳ (L.₀ c)) ⟩
R.₁ D.id
≈⟨ R.identity ⟩
≈⟨ [ R ]-resp-inverse (⊣⊢₂.unit.iso.isoʳ (L.₀ c)) ⟩
id
∎
}
Expand Down Expand Up @@ -123,20 +117,14 @@ _∘⊣⊢_ {C = C} {D} {E} {L} {R} {L′} {R′} L⊣⊢R L′⊣⊢R′ = with
; iso = λ e → record
{ isoˡ = begin
(L′.₁ (⊣⊢₁.counit.⇐.η (R′.₀ e)) ∘ ⊣⊢₂.counit.⇐.η e) ∘ ⊣⊢₂.counit.⇒.η e ∘ L′.₁ (⊣⊢₁.counit.⇒.η (R′.₀ e))
≈⟨ center (⊣⊢₂.counit.iso.isoˡ e) ⟩
L′.₁ (⊣⊢₁.counit.⇐.η (R′.₀ e)) ∘ id ∘ L′.₁ (⊣⊢₁.counit.⇒.η (R′.₀ e))
≈⟨ refl⟩∘⟨ identityˡ ⟩
≈⟨ cancelInner (⊣⊢₂.counit.iso.isoˡ e) ⟩
L′.₁ (⊣⊢₁.counit.⇐.η (R′.₀ e)) ∘ L′.₁ (⊣⊢₁.counit.⇒.η (R′.₀ e))
≈⟨ [ L′ ]-resp-∘ (⊣⊢₁.counit.iso.isoˡ (R′.₀ e)) ⟩
L′.₁ D.id
≈⟨ L′.identity ⟩
≈⟨ [ L′ ]-resp-inverse (⊣⊢₁.counit.iso.isoˡ (R′.₀ e)) ⟩
id
∎
; isoʳ = begin
(⊣⊢₂.counit.⇒.η e ∘ L′.₁ (⊣⊢₁.counit.⇒.η (R′.₀ e))) ∘ L′.₁ (⊣⊢₁.counit.⇐.η (R′.₀ e)) ∘ ⊣⊢₂.counit.⇐.η e
≈⟨ center ([ L′ ]-resp-∘ (⊣⊢₁.counit.iso.isoʳ (R′.₀ e))) ⟩
⊣⊢₂.counit.⇒.η e ∘ L′.₁ D.id ∘ ⊣⊢₂.counit.⇐.η e
≈⟨ refl⟩∘⟨ elimˡ L′.identity ⟩
≈⟨ cancelInner ([ L′ ]-resp-inverse (⊣⊢₁.counit.iso.isoʳ (R′.₀ e))) ⟩
⊣⊢₂.counit.⇒.η e ∘ ⊣⊢₂.counit.⇐.η e
≈⟨ ⊣⊢₂.counit.iso.isoʳ e ⟩
id
Expand All @@ -158,7 +146,7 @@ _∘⊣⊢_ {C = C} {D} {E} {L} {R} {L′} {R′} L⊣⊢R L′⊣⊢R′ = with
⊣⊢₂.counit.⇒.η (L′.₀ (L.₀ c)) ∘ (L′.₁ (⊣⊢₂.unit.⇒.η (L.₀ c)) ∘ L′.₁ (⊣⊢₁.counit.⇒.η (L.₀ c))) ∘ L′.₁ (L.₁ (⊣⊢₁.unit.⇒.η c))
≈⟨ pull-first ⊣⊢₂.zig ⟩
id ∘ L′.₁ (⊣⊢₁.counit.⇒.η (L.₀ c)) ∘ L′.₁ (L.₁ (⊣⊢₁.unit.⇒.η c))
≈⟨ elimʳ (([ L′ ]-resp-∘ ⊣⊢₁.zig) ○ L′.identity) ⟩
≈⟨ elimʳ ([ L′ ]-resp-inverse ⊣⊢₁.zig) ⟩
id
∎
where open E
Expand Down
6 changes: 2 additions & 4 deletions src/Categories/Category/Construction/Properties/Presheaves/FromCartesianCCC.agda
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Expand Up @@ -19,7 +19,7 @@ open import Categories.Category.Construction.Presheaves using (Presheaves′; Pr
open import Categories.Category.Instance.Setoids using (Setoids)
open import Categories.Functor.Core using (Functor)
open import Categories.Functor.Hom using (Hom[_][_,_])
open import Categories.Functor.Properties using ([_]-resp-∘; [_]-resp-square)
open import Categories.Functor.Properties using ([_]-resp-∘; [_]-resp-square; [_]-resp-inverse)
open import Categories.Functor.Presheaf using (Presheaf)
open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
open import Categories.Object.Product.Core using (Product) -- elide
Expand Down Expand Up @@ -212,9 +212,7 @@ module FromCartesianCCC {o} {C : Category o o o} (Car : Cartesian C) where
F.₁ Δ ⟨$⟩ (α.η (X × X) ⟨$⟩ (H.₁ π₁ ⟨$⟩ x , G.₁ π₂ ⟨$⟩ y))
≈⟨ α.sym-commute Δ ⟩
α.η X ⟨$⟩ (H.₁ Δ ⟨$⟩ (H.F₁ π₁ ⟨$⟩ x) , G.₁ Δ ⟨$⟩ (G.₁ π₂ ⟨$⟩ y))
≈⟨ cong (α.η X) ([ H ]-resp-∘ project₁ , [ G ]-resp-∘ project₂) ⟩
α.η X ⟨$⟩ (H.F₁ C.id ⟨$⟩ x , G.F₁ C.id ⟨$⟩ y)
≈⟨ cong (α.η X) (H.identity , G.identity) ⟩
≈⟨ cong (α.η X) ([ H ]-resp-inverse project₁ , [ G ]-resp-inverse project₂) ⟩
α.η X ⟨$⟩ (x , y)
∎ }
; curry-unique = λ {F G H} {α β} eq {X} {x} {Y} {z} →
Expand Down
4 changes: 2 additions & 2 deletions src/Categories/Diagram/Limit/Ran.agda
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Expand Up @@ -64,8 +64,8 @@ module _ {o ℓ e o′ ℓ′ e′ o″ ℓ″ e″} {C : Category o′ ℓ′ e
; sym-commute = λ _ → [ X ]-resp-square id-comm
}
; iso = λ _ → record
{ isoˡ = [ X ]-resp-∘ C.identity² ○ X.identity
; isoʳ = [ X ]-resp-∘ C.identity² ○ X.identity
{ isoˡ = [ X ]-resp-inverse C.identity²
; isoʳ = [ X ]-resp-inverse C.identity²
}
}
where open MR C using (id-comm-sym; id-comm)
Expand Down
4 changes: 2 additions & 2 deletions src/Categories/Functor/Monoidal/Properties.agda
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Expand Up @@ -210,8 +210,8 @@ module _ (A : MonoidalCategory o ℓ e) (B : MonoidalCategory o′ ℓ′ e′)
; commute = λ _ → pullʳ ([ G ]-resp-square (CF.⊗-homo.⇐.commute _)) ○ pullˡ (CG.⊗-homo.⇐.commute _) ○ C.assoc
}
; iso = λ _ → record
{ isoˡ = cancelInner ([ G ]-resp-∘ (CF.⊗-homo.iso.isoˡ _) ○ G.identity) ○ CG.⊗-homo.iso.isoˡ _
; isoʳ = cancelInner (CG.⊗-homo.iso.isoʳ _) ○ [ G ]-resp-∘ (CF.⊗-homo.iso.isoʳ _) ○ G.identity
{ isoˡ = cancelInner ([ G ]-resp-inverse (CF.⊗-homo.iso.isoˡ _)) ○ CG.⊗-homo.iso.isoˡ _
; isoʳ = cancelInner (CG.⊗-homo.iso.isoʳ _) ○ [ G ]-resp-inverse (CF.⊗-homo.iso.isoʳ _)
}
}
; associativity = ∘.associativity
Expand Down
18 changes: 9 additions & 9 deletions src/Categories/Functor/Properties.agda
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Expand Up @@ -93,19 +93,19 @@ module _ (F : Functor C D) where
D [ F₁ i ∘ F₁ g ] ∎
where open D.HomReasoning

[_]-resp-inverse : C [ C [ f ∘ g ] ≈ C.id ] → D [ D [ F₁ f ∘ F₁ g ] ≈ D.id ]
[_]-resp-inverse {f = f} {g = g} eq = begin
F₁ f D.∘ F₁ g ≈⟨ [_]-resp-∘ eq ⟩
F₁ C.id ≈⟨ identity ⟩
D.id ∎
where open D.HomReasoning

[_]-resp-Iso : Iso C f g → Iso D (F₁ f) (F₁ g)
[_]-resp-Iso {f = f} {g = g} iso = record
{ isoˡ = begin
F₁ g D.∘ F₁ f ≈⟨ [ isoˡ ]-resp-∘ ⟩
F₁ C.id ≈⟨ identity ⟩
D.id ∎
; isoʳ = begin
F₁ f D.∘ F₁ g ≈⟨ [ isoʳ ]-resp-∘ ⟩
F₁ C.id ≈⟨ identity ⟩
D.id ∎
{ isoˡ = [_]-resp-inverse isoˡ
; isoʳ = [_]-resp-inverse isoʳ
}
where open Iso iso
open D.HomReasoning

[_]-resp-≅ : F₀ Preserves _≅_ C ⟶ _≅_ D
[_]-resp-≅ i≅j = record
Expand Down