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Merge pull request #347 from agda/BaseChange
Cleaning up Slice Functor and surrounding infrastructure
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| {-# OPTIONS --without-K --safe #-} | ||
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| open import Categories.Category using (Category) | ||
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| module Categories.Adjoint.Instance.BaseChange {o ℓ e} (C : Category o ℓ e) where | ||
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| open import Categories.Adjoint using (_⊣_) | ||
| open import Categories.Adjoint.Compose using (_∘⊣_) | ||
| open import Categories.Adjoint.Instance.Slice using (Forgetful⊣Free) | ||
| open import Categories.Category.Slice C using (Slice) | ||
| open import Categories.Category.Slice.Properties C using (pullback⇒product; slice-slice≃slice) | ||
| open import Categories.Category.Equivalence.Properties using (module C≅D) | ||
| open import Categories.Diagram.Pullback C using (Pullback) | ||
| open import Categories.Functor.Slice.BaseChange C using (BaseChange!; BaseChange*) | ||
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| open Category C | ||
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| module _ {A B : Obj} (f : B ⇒ A) (pullback : ∀ {C} {h : C ⇒ A} → Pullback f h) where | ||
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| !⊣* : BaseChange! f ⊣ BaseChange* f pullback | ||
| !⊣* = C≅D.L⊣R (slice-slice≃slice f) ∘⊣ Forgetful⊣Free (Slice A) (pullback⇒product pullback) |
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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,46 @@ | ||
| {-# OPTIONS --safe --without-K #-} | ||
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| open import Categories.Category using (Category) | ||
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| module Categories.Adjoint.Instance.Slice {o ℓ e} (C : Category o ℓ e) where | ||
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| 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.NaturalTransformation using (ntHelper) | ||
| open import Categories.Object.Product C | ||
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| open Category C | ||
| open HomReasoning | ||
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| open SliceObj | ||
| open Slice⇒ | ||
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| module _ {A : Obj} (product : ∀ {X} → Product A X) where | ||
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| private | ||
| module product {X} = Product (product {X}) | ||
| open product | ||
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| Forgetful⊣Free : Forgetful ⊣ Free product | ||
| Forgetful⊣Free = record | ||
| { unit = ntHelper record | ||
| { η = λ _ → slicearr project₁ | ||
| ; commute = λ {X} {Y} f → begin | ||
| ⟨ arr Y , id ⟩ ∘ h f ≈⟨ ∘-distribʳ-⟨⟩ ⟩ | ||
| ⟨ arr Y ∘ h f , id ∘ h f ⟩ ≈⟨ ⟨⟩-cong₂ (△ f) identityˡ ⟩ | ||
| ⟨ arr X , h f ⟩ ≈˘⟨ ⟨⟩-cong₂ identityˡ identityʳ ⟩ | ||
| ⟨ id ∘ arr X , h f ∘ id ⟩ ≈˘⟨ [ product ⇒ product ]×∘⟨⟩ ⟩ | ||
| [ product ⇒ product ] id × h f ∘ ⟨ arr X , id ⟩ ∎ | ||
| } | ||
| ; counit = ntHelper record | ||
| { η = λ _ → π₂ | ||
| ; commute = λ _ → project₂ | ||
| } | ||
| ; zig = project₂ | ||
| ; zag = begin | ||
| [ product ⇒ product ]id× π₂ ∘ ⟨ π₁ , id ⟩ ≈⟨ [ product ⇒ product ]×∘⟨⟩ ⟩ | ||
| ⟨ id ∘ π₁ , π₂ ∘ id ⟩ ≈⟨ ⟨⟩-cong₂ identityˡ identityʳ ⟩ | ||
| ⟨ π₁ , π₂ ⟩ ≈⟨ η ⟩ | ||
| id ∎ | ||
| } |
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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,24 @@ | ||
| {-# OPTIONS --without-K --safe #-} | ||
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| open import Categories.Category using (Category) | ||
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| module Categories.Functor.Slice.BaseChange {o ℓ e} (C : Category o ℓ e) where | ||
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| open import Categories.Category.Slice C using (Slice) | ||
| open import Categories.Category.Slice.Properties C using (pullback⇒product; slice-slice⇒slice; slice⇒slice-slice) | ||
| open import Categories.Functor using (Functor; _∘F_) | ||
| open import Categories.Functor.Slice using (Forgetful; Free) | ||
| open import Categories.Diagram.Pullback C using (Pullback) | ||
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| open Category C | ||
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| module _ {A B : Obj} (f : B ⇒ A) where | ||
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| -- Any morphism induces a functor between slices. | ||
| BaseChange! : Functor (Slice B) (Slice A) | ||
| BaseChange! = Forgetful (Slice A) ∘F slice⇒slice-slice f | ||
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| -- Any morphism which admits pullbacks induces a functor the other way between slices. | ||
| -- This is adjoint to BaseChange!: see Categories.Adjoint.Instance.BaseChange. | ||
| BaseChange* : (∀ {C} {h : C ⇒ A} → Pullback f h) → Functor (Slice A) (Slice B) | ||
| BaseChange* pullback = slice-slice⇒slice f ∘F Free (Slice A) (pullback⇒product pullback) |