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Base change, sliced adjoints (#1120)
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| {-# OPTIONS --safe #-} | ||
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| module Cubical.Categories.Constructions.Slice.Functor where | ||
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| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Foundations.Function | ||
| open import Cubical.Foundations.Isomorphism | ||
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| open import Cubical.Categories.Category | ||
| open import Cubical.Categories.Category.Properties | ||
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| open import Cubical.Categories.Constructions.Slice.Base | ||
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| open import Cubical.Categories.Limits.Pullback | ||
| open import Cubical.Categories.Functor | ||
| open import Cubical.Categories.NaturalTransformation | ||
| open import Cubical.Categories.Adjoint | ||
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| open import Cubical.Tactics.FunctorSolver.Reflection | ||
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| open Category hiding (_∘_) | ||
| open Functor | ||
| open NatTrans | ||
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| private | ||
| variable | ||
| ℓ ℓ' : Level | ||
| C D : Category ℓ ℓ' | ||
| c d : C .ob | ||
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| infix 39 _F/_ | ||
| infix 40 ∑_ | ||
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| _F/_ : ∀ (F : Functor C D) c → Functor (SliceCat C c) (SliceCat D (F ⟅ c ⟆)) | ||
| F-ob (F F/ c) = sliceob ∘ F ⟪_⟫ ∘ S-arr | ||
| F-hom (F F/ c) h = slicehom _ | ||
| $ sym ( F-seq F _ _) ∙ cong (F ⟪_⟫) (S-comm h) | ||
| F-id (F F/ c) = SliceHom-≡-intro' _ _ $ F-id F | ||
| F-seq (F F/ c) _ _ = SliceHom-≡-intro' _ _ $ F-seq F _ _ | ||
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| ∑_ : ∀ {c d} f → Functor (SliceCat C c) (SliceCat C d) | ||
| F-ob (∑_ {C = C} f) (sliceob x) = sliceob (x ⋆⟨ C ⟩ f) | ||
| F-hom (∑_ {C = C} f) (slicehom h p) = slicehom _ $ | ||
| sym (C .⋆Assoc _ _ _) ∙ cong (comp' C f) p | ||
| F-id (∑ f) = SliceHom-≡-intro' _ _ refl | ||
| F-seq (∑ f) _ _ = SliceHom-≡-intro' _ _ refl | ||
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| module _ (Pbs : Pullbacks C) where | ||
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| open Category C using () renaming (_⋆_ to _⋆ᶜ_) | ||
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| module BaseChange {c d} (𝑓 : C [ c , d ]) where | ||
| infix 40 _＊ | ||
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| module _ {x@(sliceob arr) : SliceOb C d} where | ||
| open Pullback (Pbs (cospan _ _ _ 𝑓 arr)) public | ||
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| module _ {x} {y} ((slicehom h h-comm) : SliceCat C d [ y , x ]) where | ||
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| pbU = univProp (pbPr₁ {x = y}) (pbPr₂ ⋆ᶜ h) | ||
| (pbCommutes {x = y} ∙∙ cong (pbPr₂ ⋆ᶜ_) (sym (h-comm)) ∙∙ sym (C .⋆Assoc _ _ _)) | ||
| .fst .snd | ||
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| _＊ : Functor (SliceCat C d) (SliceCat C c) | ||
| F-ob _＊ x = sliceob (pbPr₁ {x = x}) | ||
| F-hom _＊ f = slicehom _ (sym (fst (pbU f))) | ||
| F-id _＊ = SliceHom-≡-intro' _ _ $ pullbackArrowUnique (sym (C .⋆IdL _)) (C .⋆IdR _ ∙ sym (C .⋆IdL _)) | ||
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| F-seq _＊ _ _ = | ||
| let (u₁ , v₁) = pbU _ ; (u₂ , v₂) = pbU _ | ||
| in SliceHom-≡-intro' _ _ $ pullbackArrowUnique | ||
| (u₂ ∙∙ cong (C ⋆ _) u₁ ∙∙ sym (C .⋆Assoc _ _ _)) | ||
| (sym (C .⋆Assoc _ _ _) ∙∙ cong (comp' C _) v₂ ∙∙ AssocCong₂⋆R C v₁) | ||
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| open BaseChange using (_＊) | ||
| open NaturalBijection renaming (_⊣_ to _⊣₂_) | ||
| open Iso | ||
| open _⊣₂_ | ||
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| module _ (𝑓 : C [ c , d ]) where | ||
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| open BaseChange 𝑓 hiding (_＊) | ||
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| ∑𝑓⊣𝑓＊ : ∑ 𝑓 ⊣₂ 𝑓 ＊ | ||
| fun (adjIso ∑𝑓⊣𝑓＊) (slicehom h o) = | ||
| let ((_ , (p , _)) , _) = univProp _ _ (sym o) | ||
| in slicehom _ (sym p) | ||
| inv (adjIso ∑𝑓⊣𝑓＊) (slicehom h o) = slicehom _ $ | ||
| AssocCong₂⋆R C (sym (pbCommutes)) ∙ cong (_⋆ᶜ 𝑓) o | ||
| rightInv (adjIso ∑𝑓⊣𝑓＊) (slicehom h o) = | ||
| SliceHom-≡-intro' _ _ (pullbackArrowUnique (sym o) refl) | ||
| leftInv (adjIso ∑𝑓⊣𝑓＊) (slicehom h o) = | ||
| let ((_ , (_ , q)) , _) = univProp _ _ _ | ||
| in SliceHom-≡-intro' _ _ (sym q) | ||
| adjNatInD ∑𝑓⊣𝑓＊ f k = SliceHom-≡-intro' _ _ $ | ||
| let ((h' , (v' , u')) , _) = univProp _ _ _ | ||
| ((_ , (v'' , u'')) , _) = univProp _ _ _ | ||
| in pullbackArrowUnique (v' ∙∙ cong (h' ⋆ᶜ_) v'' ∙∙ sym (C .⋆Assoc _ _ _)) | ||
| (cong (_⋆ᶜ _) u' ∙ AssocCong₂⋆R C u'') | ||
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| adjNatInC ∑𝑓⊣𝑓＊ g h = SliceHom-≡-intro' _ _ $ C .⋆Assoc _ _ _ | ||
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| open UnitCounit | ||
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| module SlicedAdjoint {L : Functor C D} {R} (L⊣R : L ⊣ R) where | ||
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| open Category D using () renaming (_⋆_ to _⋆ᵈ_) | ||
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| open _⊣_ L⊣R | ||
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| module _ {c} {d} where | ||
| module aI = Iso (adjIso (adj→adj' _ _ L⊣R) {c} {d}) | ||
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| module Left (b : D .ob) where | ||
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| ⊣F/ : Functor (SliceCat C (R ⟅ b ⟆)) (SliceCat D b) | ||
| ⊣F/ = ∑ (ε ⟦ b ⟧) ∘F L F/ (R ⟅ b ⟆) | ||
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| L/b⊣R/b : ⊣F/ ⊣₂ (R F/ b) | ||
| fun (adjIso L/b⊣R/b) (slicehom _ p) = slicehom _ $ | ||
| C .⋆Assoc _ _ _ | ||
| ∙∙ cong (_ ⋆ᶜ_) (sym (F-seq R _ _) ∙∙ cong (R ⟪_⟫) p ∙∙ F-seq R _ _) | ||
| ∙∙ AssocCong₂⋆L C (sym (N-hom η _)) | ||
| ∙∙ cong (_ ⋆ᶜ_) (Δ₂ _) | ||
| ∙∙ C .⋆IdR _ | ||
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| inv (adjIso L/b⊣R/b) (slicehom _ p) = | ||
| slicehom _ $ AssocCong₂⋆R D (sym (N-hom ε _)) | ||
| ∙ cong (_⋆ᵈ _) (sym (F-seq L _ _) ∙ cong (L ⟪_⟫) p) | ||
| rightInv (adjIso L/b⊣R/b) _ = SliceHom-≡-intro' _ _ $ aI.rightInv _ | ||
| leftInv (adjIso L/b⊣R/b) _ = SliceHom-≡-intro' _ _ $ aI.leftInv _ | ||
| adjNatInD L/b⊣R/b _ _ = SliceHom-≡-intro' _ _ $ | ||
| cong (_ ⋆ᶜ_) (F-seq R _ _) ∙ sym (C .⋆Assoc _ _ _) | ||
| adjNatInC L/b⊣R/b _ _ = SliceHom-≡-intro' _ _ $ | ||
| cong (_⋆ᵈ _) (F-seq L _ _) ∙ (D .⋆Assoc _ _ _) | ||
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| module Right (b : C .ob) where | ||
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| F/⊣ : Functor (SliceCat D (L ⟅ b ⟆)) (SliceCat C b) | ||
| F/⊣ = (η ⟦ b ⟧) ＊ ∘F R F/ (L ⟅ b ⟆) | ||
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| open BaseChange (η ⟦ b ⟧) hiding (_＊) | ||
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| L/b⊣R/b : L F/ b ⊣₂ F/⊣ | ||
| fun (adjIso L/b⊣R/b) (slicehom f s) = slicehom _ $ | ||
| sym $ univProp _ _ (N-hom η _ ∙∙ | ||
| cong (_ ⋆ᶜ_) (cong (R ⟪_⟫) (sym s) ∙ F-seq R _ _) | ||
| ∙∙ sym (C .⋆Assoc _ _ _)) .fst .snd .fst | ||
| inv (adjIso L/b⊣R/b) (slicehom f s) = slicehom _ | ||
| (D .⋆Assoc _ _ _ | ||
| ∙∙ congS (_⋆ᵈ (ε ⟦ _ ⟧ ⋆⟨ D ⟩ _)) (F-seq L _ _) | ||
| ∙∙ D .⋆Assoc _ _ _ ∙ cong (L ⟪ f ⟫ ⋆ᵈ_) | ||
| (cong (L ⟪ pbPr₂ ⟫ ⋆ᵈ_) (sym (N-hom ε _)) | ||
| ∙∙ sym (D .⋆Assoc _ _ _) | ||
| ∙∙ cong (_⋆ᵈ ε ⟦ F-ob L b ⟧) | ||
| (preserveCommF L $ sym pbCommutes) | ||
| ∙∙ D .⋆Assoc _ _ _ | ||
| ∙∙ cong (L ⟪ pbPr₁ ⟫ ⋆ᵈ_) (Δ₁ b) | ||
| ∙ D .⋆IdR _) | ||
| ∙∙ sym (F-seq L _ _) | ||
| ∙∙ cong (L ⟪_⟫) s) | ||
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| rightInv (adjIso L/b⊣R/b) h = SliceHom-≡-intro' _ _ $ | ||
| let p₂ : ∀ {x} → η ⟦ _ ⟧ ⋆ᶜ R ⟪ L ⟪ x ⟫ ⋆⟨ D ⟩ ε ⟦ _ ⟧ ⟫ ≡ x | ||
| p₂ = cong (_ ⋆ᶜ_) (F-seq R _ _) ∙ | ||
| AssocCong₂⋆L C (sym (N-hom η _)) | ||
| ∙∙ cong (_ ⋆ᶜ_) (Δ₂ _) ∙∙ C .⋆IdR _ | ||
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| in pullbackArrowUnique (sym (S-comm h)) p₂ | ||
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| leftInv (adjIso L/b⊣R/b) _ = SliceHom-≡-intro' _ _ $ | ||
| cong ((_⋆ᵈ _) ∘ L ⟪_⟫) (sym (snd (snd (fst (univProp _ _ _))))) | ||
| ∙ aI.leftInv _ | ||
| adjNatInD L/b⊣R/b _ _ = SliceHom-≡-intro' _ _ $ | ||
| let (h , (u , v)) = univProp _ _ _ .fst | ||
| (u' , v') = pbU _ | ||
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| in pullbackArrowUnique | ||
| (u ∙∙ cong (h ⋆ᶜ_) u' ∙∙ sym (C .⋆Assoc h _ _)) | ||
| (cong (_ ⋆ᶜ_) (F-seq R _ _) | ||
| ∙∙ sym (C .⋆Assoc _ _ _) ∙∙ | ||
| (cong (_⋆ᶜ _) v ∙ AssocCong₂⋆R C v')) | ||
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| adjNatInC L/b⊣R/b g h = let w = _ in SliceHom-≡-intro' _ _ $ | ||
| cong (_⋆ᵈ _) (cong (L ⟪_⟫) (C .⋆Assoc _ _ w) ∙ F-seq L _ (_ ⋆ᶜ w)) | ||
| ∙ D .⋆Assoc _ _ _ | ||
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