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| {-# OPTIONS --allow-unsolved-metas #-} | ||
| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Categories.Category | ||
| open import Cubical.Categories.Functor | ||
| open import Cubical.Categories.Limits.Pullback | ||
| open import Cubical.Categories.Constructions.Slice | ||
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| module Categories.PullbackFunctor where | ||
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| private | ||
| variable | ||
| ℓ ℓ' : Level | ||
| module _ (C : Category ℓ ℓ') (pullbacks : Pullbacks C) where | ||
| open Category C | ||
| open Pullback | ||
| open SliceOb | ||
| open SliceHom | ||
| module _ {X Y : ob} (f : Hom[ X , Y ]) where | ||
| module TransformMaps {A B : ob} (m : Hom[ A , Y ]) (n : Hom[ B , Y ]) (k : Hom[ A , B ]) (tri : k ⋆ n ≡ m) where | ||
| cospanA : Cospan C | ||
| cospanA = cospan X Y A f m | ||
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| cospanB : Cospan C | ||
| cospanB = cospan X Y B f n | ||
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| P : ob | ||
| P = pullbacks cospanA .pbOb | ||
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| Q : ob | ||
| Q = pullbacks cospanB .pbOb | ||
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| f*m : Hom[ P , X ] | ||
| f*m = pullbacks cospanA .pbPr₁ | ||
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| g : Hom[ P , A ] | ||
| g = pullbacks cospanA .pbPr₂ | ||
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| f*n : Hom[ Q , X ] | ||
| f*n = pullbacks cospanB .pbPr₁ | ||
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| h : Hom[ Q , B ] | ||
| h = pullbacks cospanB .pbPr₂ | ||
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| f*m⋆f≡g⋆m : f*m ⋆ f ≡ g ⋆ m | ||
| f*m⋆f≡g⋆m = pullbacks cospanA .pbCommutes | ||
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| g⋆k : Hom[ P , B ] | ||
| g⋆k = g ⋆ k | ||
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| g⋆k⋆n≡f*m⋆f : (g ⋆ k) ⋆ n ≡ f*m ⋆ f | ||
| g⋆k⋆n≡f*m⋆f = | ||
| (g ⋆ k) ⋆ n | ||
| ≡⟨ ⋆Assoc g k n ⟩ | ||
| g ⋆ (k ⋆ n) | ||
| ≡⟨ cong (λ x → g ⋆ x) tri ⟩ | ||
| g ⋆ m | ||
| ≡⟨ sym f*m⋆f≡g⋆m ⟩ | ||
| f*m ⋆ f | ||
| ∎ | ||
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| arrow : Hom[ P , Q ] | ||
| arrow = pullbackArrow C (pullbacks cospanB) f*m g⋆k (sym g⋆k⋆n≡f*m⋆f) | ||
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| arrow⋆f*n≡f*m : arrow ⋆ f*n ≡ f*m | ||
| arrow⋆f*n≡f*m = sym (pullbackArrowPr₁ C (pullbacks cospanB) f*m g⋆k (sym g⋆k⋆n≡f*m⋆f)) | ||
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| reindexFunctor : Functor (SliceCat C Y) (SliceCat C X) | ||
| Functor.F-ob reindexFunctor (sliceob {A} m) = sliceob (pullbacks (cospan X Y A f m) .pbPr₁) | ||
| Functor.F-hom reindexFunctor {sliceob {A} m} {sliceob {B} n} (slicehom k tri) = slicehom arrow arrow⋆f*n≡f*m where open TransformMaps m n k tri | ||
| Functor.F-id reindexFunctor {sliceob {A} m} = SliceHom-≡-intro' C X (pullbackArrowUnique C (pullbacks cospanB) f*m g⋆k (sym g⋆k⋆n≡f*m⋆f) id (sym (⋆IdL f*m)) (⋆IdR g ∙ sym (⋆IdL g))) where open TransformMaps m m id (⋆IdL m) | ||
| Functor.F-seq reindexFunctor {sliceob {A} m} {sliceob {B} n} {sliceob {C'} o} (slicehom j jComm) (slicehom k kComm) = SliceHom-≡-intro' C X {!!} where | ||
| module ABTransform = TransformMaps m n j jComm | ||
| module BCTransform = TransformMaps n o k kComm | ||
| module ACTransform = TransformMaps m o (j ⋆ k) (⋆Assoc j k o ∙ cong (λ x → j ⋆ x) kComm ∙ jComm) | ||
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| P : ob | ||
| P = ABTransform.P | ||
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| f*m : Hom[ P , X ] | ||
| f*m = ABTransform.f*m | ||
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| Q : ob | ||
| Q = ABTransform.Q | ||
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| R : ob | ||
| R = BCTransform.P | ||
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| f*n : Hom[ Q , X ] | ||
| f*n = ABTransform.f*n | ||
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| g : Hom[ P , A ] | ||
| g = ABTransform.g | ||
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| f*m⋆f≡g⋆m : f*m ⋆ f ≡ g ⋆ m | ||
| f*m⋆f≡g⋆m = ABTransform.f*m⋆f≡g⋆m | ||
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| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Foundations.HLevels | ||
| open import Cubical.Foundations.Equiv | ||
| open import Cubical.Foundations.Isomorphism | ||
| open import Cubical.Data.Sigma | ||
| open import Cubical.Data.FinData | ||
| open import Cubical.HITs.PropositionalTruncation as PT hiding (map) | ||
| open import Cubical.HITs.PropositionalTruncation.Monad | ||
| open import Cubical.Categories.Limits.Pullback | ||
| open import Cubical.Categories.Category.Base | ||
| open import Cubical.Categories.Functor | ||
| open import Cubical.Categories.Constructions.Slice | ||
| open import Cubical.Categories.Adjoint | ||
| open import Realizability.CombinatoryAlgebra | ||
| open import Realizability.ApplicativeStructure | ||
| open import Categories.PullbackFunctor | ||
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| module Realizability.Assembly.LocallyCartesianClosed {ℓ} {A : Type ℓ} (ca : CombinatoryAlgebra A) where | ||
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| open CombinatoryAlgebra ca | ||
| open Realizability.CombinatoryAlgebra.Combinators ca renaming (i to Id; ia≡a to Ida≡a) hiding (Z) | ||
| open import Realizability.Assembly.Base ca | ||
| open import Realizability.Assembly.Morphism ca | ||
| open import Realizability.Assembly.Pullbacks ca | ||
| open import Realizability.Assembly.Reindexing ca | ||
| open NaturalBijection | ||
| open SliceOb | ||
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| module _ {X Y : Type ℓ} {asmX : Assembly X} {asmY : Assembly Y} (f : AssemblyMorphism asmX asmY) where | ||
| module SliceDomain {V : Type ℓ} {asmV : Assembly V} (h : AssemblyMorphism asmV asmX) where | ||
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| D : Type ℓ | ||
| D = Σ[ y ∈ Y ] Σ[ s ∈ (fiber (f .map) y → V)] (∀ (yFiberF : fiber (f .map) y) → h .map (s yFiberF) ≡ yFiberF .fst) | ||
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| isSetD : isSet D | ||
| isSetD = | ||
| isSetΣ | ||
| (asmY .isSetX) | ||
| (λ y → | ||
| isSetΣ | ||
| (isSet→ (asmV .isSetX)) | ||
| (λ s → | ||
| isSetΠ λ yFiberF → isProp→isSet (asmX .isSetX _ _))) | ||
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| _⊩D_ : A → D → Type ℓ | ||
| r ⊩D (y , s , coh) = ((pr₁ ⨾ r) ⊩[ asmY ] y) × (∀ (yFiberF : fiber (f .map) y) (a : A) → a ⊩[ asmX ] (yFiberF .fst) → (pr₂ ⨾ r ⨾ a) ⊩[ asmV ] (s yFiberF)) | ||
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| isProp⊩D : ∀ r d → isProp (r ⊩D d) | ||
| isProp⊩D r d = | ||
| isProp× | ||
| (asmY .⊩isPropValued _ _) | ||
| (isPropΠ | ||
| λ yFiberF → | ||
| isPropΠ | ||
| λ a → | ||
| isProp→ (asmV .⊩isPropValued _ _)) | ||
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| asmD : Assembly (Σ[ d ∈ D ] ∃[ r ∈ A ] (r ⊩D d)) | ||
| Assembly._⊩_ asmD r (d@(y , s , coh) , ∃r) = r ⊩D d | ||
| Assembly.isSetX asmD = isSetΣ isSetD (λ d → isProp→isSet isPropPropTrunc) | ||
| Assembly.⊩isPropValued asmD r (d@(y , s , coh) , ∃a) = isProp⊩D r d | ||
| Assembly.⊩surjective asmD (d , ∃a) = ∃a | ||
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| projection : AssemblyMorphism asmD asmY | ||
| AssemblyMorphism.map projection (d@(y , s , coh) , ∃r) = y | ||
| AssemblyMorphism.tracker projection = | ||
| return | ||
| (pr₁ , | ||
| (λ { (d@(y , s , coh) , ∃a) r r⊩d → r⊩d .fst })) | ||
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| private module SliceDomainHom {V W : Type ℓ} {asmV : Assembly V} {asmW : Assembly W} (g : AssemblyMorphism asmV asmX) (h : AssemblyMorphism asmW asmX) (j : AssemblyMorphism asmV asmW) (comm : j ⊚ h ≡ g) where | ||
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| rawMap : SliceDomain.D g → SliceDomain.D h | ||
| rawMap (y , s , coh) = | ||
| y , | ||
| (λ fib → j .map (s fib)) , | ||
| λ { fib@(x , fx≡y) → | ||
| h .map (j .map (s fib)) | ||
| ≡[ i ]⟨ comm i .map (s fib) ⟩ | ||
| g .map (s fib) | ||
| ≡⟨ coh fib ⟩ | ||
| x | ||
| ∎ } | ||
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| transformRealizers : ∀ (d : SliceDomain.D g) → Σ[ b ∈ A ] (SliceDomain._⊩D_ g b d) → Σ[ j~ ∈ A ] (tracks {xs = asmV} {ys = asmW} j~ (j .map)) → Σ[ r ∈ A ] (SliceDomain._⊩D_ h r (rawMap d)) | ||
| transformRealizers d@(y , s , coh) (e , pr₁e⊩y , pr₂e⊩) (j~ , isTrackedJ) = | ||
| let | ||
| realizer : A | ||
| realizer = pair ⨾ (pr₁ ⨾ e) ⨾ λ* (` j~ ̇ (` pr₂ ̇ ` e ̇ # zero)) | ||
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| pr₁realizer≡pr₁e : pr₁ ⨾ realizer ≡ pr₁ ⨾ e | ||
| pr₁realizer≡pr₁e = | ||
| pr₁ ⨾ realizer | ||
| ≡⟨ refl ⟩ -- unfolding | ||
| pr₁ ⨾ (pair ⨾ (pr₁ ⨾ e) ⨾ λ* (` j~ ̇ (` pr₂ ̇ ` e ̇ # zero))) | ||
| ≡⟨ pr₁pxy≡x _ _ ⟩ | ||
| pr₁ ⨾ e | ||
| ∎ | ||
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| pr₂realizerEq : (a : A) → pr₂ ⨾ realizer ⨾ a ≡ j~ ⨾ (pr₂ ⨾ e ⨾ a) | ||
| pr₂realizerEq a = | ||
| pr₂ ⨾ realizer ⨾ a | ||
| ≡⟨ refl ⟩ | ||
| pr₂ ⨾ (pair ⨾ (pr₁ ⨾ e) ⨾ λ* (` j~ ̇ (` pr₂ ̇ ` e ̇ # zero))) ⨾ a | ||
| ≡⟨ cong (λ x → x ⨾ a) (pr₂pxy≡y _ _) ⟩ | ||
| λ* (` j~ ̇ (` pr₂ ̇ ` e ̇ # zero)) ⨾ a | ||
| ≡⟨ λ*ComputationRule (` j~ ̇ (` pr₂ ̇ ` e ̇ # zero)) a ⟩ | ||
| j~ ⨾ (pr₂ ⨾ e ⨾ a) | ||
| ∎ | ||
| in | ||
| (realizer , | ||
| (subst (λ r' → r' ⊩[ asmY ] y) (sym pr₁realizer≡pr₁e) pr₁e⊩y , | ||
| (λ { fib@(x , fx≡y) a a⊩x → | ||
| subst (λ r' → r' ⊩[ asmW ] (j .map (s fib))) (sym (pr₂realizerEq a)) (isTrackedJ (s fib) (pr₂ ⨾ e ⨾ a) (pr₂e⊩ fib a a⊩x)) }))) | ||
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| morphism : AssemblyMorphism (SliceDomain.asmD g) (SliceDomain.asmD h) | ||
| AssemblyMorphism.map morphism (d@(y , s , coh) , ∃r) = rawMap d , PT.rec2 isPropPropTrunc (λ r⊩d j~ → ∣ transformRealizers d r⊩d j~ ∣₁) ∃r (j .tracker) | ||
| AssemblyMorphism.tracker morphism = | ||
| do | ||
| (j~ , isTrackedJ) ← j .tracker | ||
| let | ||
| closure : Term as 2 | ||
| closure = (` j~ ̇ (` pr₂ ̇ # one ̇ # zero)) | ||
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| realizer : Term as 1 | ||
| realizer = ` pair ̇ (` pr₁ ̇ # zero) ̇ (λ*abst closure) | ||
| return | ||
| (λ* realizer , | ||
| (λ { (d@(y , s , coh) , ∃r) a (pr₁a⊩y , pr₂a⊩) → | ||
| let | ||
| pr₁Eq : pr₁ ⨾ (λ* realizer ⨾ a) ≡ pr₁ ⨾ a | ||
| pr₁Eq = | ||
| pr₁ ⨾ (λ* realizer ⨾ a) | ||
| ≡⟨ cong (λ x → pr₁ ⨾ x) (λ*ComputationRule realizer a) ⟩ | ||
| pr₁ ⨾ (pair ⨾ (pr₁ ⨾ a) ⨾ _) | ||
| ≡⟨ pr₁pxy≡x _ _ ⟩ | ||
| pr₁ ⨾ a | ||
| ∎ | ||
| in | ||
| subst (λ r' → r' ⊩[ asmY ] y) (sym pr₁Eq) pr₁a⊩y , | ||
| (λ { fib@(x , fx≡y) b b⊩x → {!isTrackedJ !} }) })) | ||
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| Π[_] : Functor (SliceCat ASM (X , asmX)) (SliceCat ASM (Y , asmY)) | ||
| Functor.F-ob Π[_] (sliceob {V , asmV} h) = sliceob (SliceDomain.projection h) | ||
| Functor.F-hom Π[_] {sliceob {V , asmV} g} {sliceob {W , asmW} h} (slicehom j comm) = {!!} | ||
| Functor.F-id Π[_] = {!!} | ||
| Functor.F-seq Π[_] = {!!} | ||
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| reindex⊣Π[_] : reindexFunctor ASM PullbacksASM f ⊣ Π[_] | ||
| Iso.fun (_⊣_.adjIso reindex⊣Π[_]) = λ fo → slicehom {!!} {!!} | ||
| Iso.inv (_⊣_.adjIso reindex⊣Π[_]) = {!!} | ||
| Iso.rightInv (_⊣_.adjIso reindex⊣Π[_]) = {!!} | ||
| Iso.leftInv (_⊣_.adjIso reindex⊣Π[_]) = {!!} | ||
| _⊣_.adjNatInD reindex⊣Π[_] = {!!} | ||
| _⊣_.adjNatInC reindex⊣Π[_] = {!!} |
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