fix(hott/algebra) fix some proofs for functor category
This commit is contained in:
Jakob von Raumer
Leonardo de Moura
parent
31b3bbe5cb
commit
31d1076bd7
2 changed files with 143 additions and 82 deletions
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@ -2,9 +2,10 @@
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Authors: Floris van Doorn, Jakob von Raumer
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import .basic
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import .basic types.pi
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open function precategory eq prod equiv is_equiv sigma sigma.ops
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open function precategory eq prod equiv is_equiv sigma sigma.ops truncation
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open pi
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structure functor (C D : Precategory) : Type :=
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(obF : C → D)
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@ -36,20 +37,22 @@ namespace functor
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exact (pr₁ S.2.2), exact (pr₂ S.2.2),
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fapply adjointify,
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intro F, apply (functor.rec_on F), intros (d1, d2, d3, d4),
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exact (dpair d1 (dpair d2 (pair d3 (@d4)))),
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exact (sigma.mk d1 (sigma.mk d2 (pair d3 (@d4)))),
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intro F, apply (functor.rec_on F), intros (d1, d2, d3, d4), apply idp,
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intro S, apply (sigma.rec_on S), intros (d1, S2),
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apply (sigma.rec_on S2), intros (d2, P1),
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apply (prod.rec_on P1), intros (d3, d4), apply idp,
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end
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-- The following lemmas will later be used to prove that the type of
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-- precategories formes a precategory itself
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protected definition compose (G : functor D E) (F : functor C D) : functor C E :=
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functor.mk
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(λx, G (F x))
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(λ x, G (F x))
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(λ a b f, G (F f))
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(λ a, calc
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G (F (ID a)) = G id : {respect_id F a} --not giving the braces explicitly makes the elaborator compute a couple more seconds
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... = id : respect_id G (F a))
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G (F (ID a)) = G (ID (F a)) : {respect_id F a}
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... = ID (G (F a)) : respect_id G (F a))
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(λ a b c g f, calc
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G (F (g ∘ f)) = G (F g ∘ F f) : respect_comp F g f
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... = G (F g) ∘ G (F f) : respect_comp G (F g) (F f))
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@ -57,38 +60,87 @@ namespace functor
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infixr `∘f`:60 := compose
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protected theorem congr
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{C : Precategory} {D : Precategory}
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(F : C → D)
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(foo2 : Π ⦃a b : C⦄, hom a b → hom (F a) (F b))
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(foo3a foo3b : Π (a : C), foo2 (ID a) = ID (F a))
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(foo4a foo4b : Π {a b c : C} (g : @hom C C b c) (f : @hom C C a b),
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foo2 (g ∘ f) = foo2 g ∘ foo2 f)
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(p3 : foo3a = foo3b) (p4 : @foo4a = @foo4b)
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: functor.mk F foo2 foo3a @foo4a = functor.mk F foo2 foo3b @foo4b
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:=
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begin
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apply (eq.rec_on p3), intros,
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apply (eq.rec_on p4), intros,
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apply idp,
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end
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protected theorem assoc {A B C D : Precategory} (H : functor C D) (G : functor B C) (F : functor A B) :
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H ∘f (G ∘f F) = (H ∘f G) ∘f F :=
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sorry
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begin
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apply (functor.rec_on H), intros (H1, H2, H3, H4),
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apply (functor.rec_on G), intros (G1, G2, G3, G4),
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apply (functor.rec_on F), intros (F1, F2, F3, F4),
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fapply functor.congr,
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apply funext.path_pi, intro a,
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apply (@is_hset.elim), apply !homH,
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apply funext.path_pi, intro a,
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repeat (apply funext.path_pi; intros),
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apply (@is_hset.elim), apply !homH,
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end
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/-protected definition id {C : Precategory} : functor C C :=
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protected definition id {C : Precategory} : functor C C :=
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mk (λa, a) (λ a b f, f) (λ a, idp) (λ a b c f g, idp)
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protected definition ID (C : Precategory) : functor C C := id
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protected theorem id_left (F : functor C D) : id ∘f F = F :=
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functor.rec (λ obF homF idF compF, dcongr_arg4 mk idp idp !proof_irrel !proof_irrel) F
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begin
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apply (functor.rec_on F), intros (F1, F2, F3, F4),
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fapply functor.congr,
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apply funext.path_pi, intro a,
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apply (@is_hset.elim), apply !homH,
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repeat (apply funext.path_pi; intros),
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apply (@is_hset.elim), apply !homH,
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end
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protected theorem id_right (F : functor C D) : F ∘f id = F :=
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functor.rec (λ obF homF idF compF, dcongr_arg4 mk idp idp !proof_irrel !proof_irrel) F-/
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begin
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apply (functor.rec_on F), intros (F1, F2, F3, F4),
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fapply functor.congr,
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apply funext.path_pi, intro a,
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apply (@is_hset.elim), apply !homH,
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repeat (apply funext.path_pi; intros),
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apply (@is_hset.elim), apply !homH,
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end
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end functor
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/-
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namespace category
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open functor
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definition category_of_categories [reducible] : category Category :=
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definition precategory_of_precategories : precategory Precategory :=
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mk (λ a b, functor a b)
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sorry -- TODO: Show that functors form a set?
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(λ a b c g f, functor.compose g f)
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(λ a, functor.id)
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(λ a b c d h g f, !functor.assoc)
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(λ a b f, !functor.id_left)
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(λ a b f, !functor.id_right)
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definition Category_of_categories [reducible] := Mk category_of_categories
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definition Precategory_of_categories := Mk precategory_of_precategories
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namespace ops
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notation `Cat`:max := Category_of_categories
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instance [persistent] category_of_categories
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notation `PreCat`:max := Precategory_of_categories
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instance [persistent] precategory_of_precategories
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end ops
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end category-/
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end category
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namespace functor
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-- open category.ops
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@ -15,33 +15,12 @@ infixl `⟹`:25 := natural_transformation -- \==>
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namespace natural_transformation
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variables {C D : Precategory} {F G H I : functor C D}
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definition natural_map [coercion] (η : F ⟹ G) : Π(a : C), F a ⟶ G a :=
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definition natural_map [coercion] (η : F ⟹ G) : Π (a : C), F a ⟶ G a :=
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rec (λ x y, x) η
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theorem naturality (η : F ⟹ G) : Π⦃a b : C⦄ (f : a ⟶ b), G f ∘ η a = η b ∘ F f :=
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rec (λ x y, y) η
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protected definition sigma_char :
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(Σ (η : Π (a : C), hom (F a) (G a)), Π (a b : C) (f : hom a b), G f ∘ η a = η b ∘ F f) ≃ (F ⟹ G) :=
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/-equiv.mk (λ S, natural_transformation.mk S.1 S.2)
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(adjointify (λ S, natural_transformation.mk S.1 S.2)
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(λ H, natural_transformation.rec_on H (λ η nat, dpair η nat))
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(λ H, natural_transformation.rec_on H (λ η nat, idpath (natural_transformation.mk η nat)))
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(λ S, sigma.rec_on S (λ η nat, idpath (dpair η nat))))-/
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/- THE FOLLLOWING CAUSES LEAN TO SEGFAULT?
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begin
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fapply equiv.mk,
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intro S, apply natural_transformation.mk, exact (S.2),
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fapply adjointify,
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intro H, apply (natural_transformation.rec_on H), intros (η, natu),
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exact (dpair η @natu),
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intro H, apply (natural_transformation.rec_on _ _ _),
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intros,
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end
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check sigma_char-/
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sorry
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protected definition compose (η : G ⟹ H) (θ : F ⟹ G) : F ⟹ H :=
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natural_transformation.mk
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(λ a, η a ∘ θ a)
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@ -52,64 +31,94 @@ namespace natural_transformation
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... = η b ∘ (G f ∘ θ a) : assoc
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... = η b ∘ (θ b ∘ F f) : naturality θ f
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... = (η b ∘ θ b) ∘ F f : assoc)
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--congr_arg (λx, η b ∘ x) (naturality θ f) -- this needed to be explicit for some reason (on Oct 24)
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infixr `∘n`:60 := compose
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protected definition assoc (η₃ : H ⟹ I) (η₂ : G ⟹ H) (η₁ : F ⟹ G) [fext fext2 fext3 : funext] :
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η₃ ∘n (η₂ ∘n η₁) = (η₃ ∘n η₂) ∘n η₁ :=
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-- Proof broken, universe issues?
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/-have aux [visible] : is_hprop (Π (a b : C) (f : hom a b), I f ∘ (η₃ ∘n η₂) a ∘ η₁ a = ((η₃ ∘n η₂) b ∘ η₁ b) ∘ F f),
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protected theorem congr
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{C : Precategory} {D : Precategory}
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(F G : C ⇒ D)
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(η₁ η₂ : Π (a : C), hom (F a) (G a))
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(nat₁ : Π (a b : C) (f : hom a b), G f ∘ η₁ a = η₁ b ∘ F f)
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(nat₂ : Π (a b : C) (f : hom a b), G f ∘ η₂ a = η₂ b ∘ F f)
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(p₁ : η₁ = η₂) (p₂ : p₁ ▹ nat₁ = nat₂)
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: @natural_transformation.mk C D F G η₁ nat₁ = @natural_transformation.mk C D F G η₂ nat₂
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:=
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begin
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repeat (apply trunc_pi; intros),
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apply (succ_is_trunc -1 (I a_2 ∘ (η₃ ∘n η₂) a ∘ η₁ a)),
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end,
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dcongr_arg2 mk (funext.path_forall _ _ (λ x, !assoc)) !is_hprop.elim-/
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sorry
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apply (dcongr_arg2 (@natural_transformation.mk C D F G) p₁ p₂),
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end
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protected definition assoc (η₃ : H ⟹ I) (η₂ : G ⟹ H) (η₁ : F ⟹ G) :
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η₃ ∘n (η₂ ∘n η₁) = (η₃ ∘n η₂) ∘n η₁ :=
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begin
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apply (rec_on η₃), intros (η₃1, η₃2),
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apply (rec_on η₂), intros (η₂1, η₂2),
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apply (rec_on η₁), intros (η₁1, η₁2),
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fapply natural_transformation.congr,
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apply funext.path_pi, intro a,
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apply assoc,
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apply funext.path_pi, intro a,
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apply funext.path_pi, intro b,
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apply funext.path_pi, intro f,
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apply (@is_hset.elim), apply !homH,
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end
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protected definition id {C D : Precategory} {F : functor C D} : natural_transformation F F :=
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mk (λa, id) (λa b f, !id_right ⬝ (!id_left⁻¹))
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protected definition ID {C D : Precategory} (F : functor C D) : natural_transformation F F := id
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protected definition id_left (η : F ⟹ G) [fext : funext.{l_1 l_4}] :
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id ∘n η = η :=
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--Proof broken like all trunc_pi proofs
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protected definition ID {C D : Precategory} (F : functor C D) : natural_transformation F F :=
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id
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protected definition id_left (η : F ⟹ G) : id ∘n η = η :=
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begin
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apply (rec_on η), intros (η₁, nat₁),
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fapply (natural_transformation.congr F G),
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apply funext.path_pi, intro a,
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apply id_left,
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apply funext.path_pi, intro a,
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apply funext.path_pi, intro b,
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apply funext.path_pi, intro f,
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apply (@is_hset.elim), apply !homH,
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end
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protected definition id_right (η : F ⟹ G) : η ∘n id = η :=
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begin
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apply (rec_on η), intros (η₁, nat₁),
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fapply (natural_transformation.congr F G),
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apply funext.path_pi, intro a,
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apply id_right,
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apply funext.path_pi, intro a,
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apply funext.path_pi, intro b,
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apply funext.path_pi, intro f,
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apply (@is_hset.elim), apply !homH,
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end
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protected definition sigma_char :
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(Σ (η : Π (a : C), hom (F a) (G a)), Π (a b : C) (f : hom a b), G f ∘ η a = η b ∘ F f) ≃ (F ⟹ G) :=
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/-begin
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apply (rec_on η), intros (f, H),
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fapply (path.dcongr_arg2 mk),
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apply (funext.path_forall _ f (λa, !id_left)),
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assert (H1 : is_hprop (Π {a b : C} (g : hom a b), G g ∘ f a = f b ∘ F g)),
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--repeat (apply trunc_pi; intros),
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apply (@trunc_pi _ _ _ (-2 .+1) _),
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/- apply (succ_is_trunc -1 (G a_2 ∘ f a) (f a_1 ∘ F a_2)),
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apply (!is_hprop.elim),-/
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intro what,
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fapply equiv.mk,
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intro S, apply natural_transformation.mk, exact (S.2),
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fapply adjointify,
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intro H, apply (natural_transformation.rec_on H), intros (η, natu),
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exact (sigma.mk η @natu),
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intro H, apply (natural_transformation.rec_on _ _ _),
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intro S2,
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end-/
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/-(λ x, equiv.mk (λ S, natural_transformation.mk S.1 S.2)
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(adjointify (λ S, natural_transformation.mk S.1 S.2)
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(λ H, natural_transformation.rec_on H (λ η nat, sigma.mk η nat))
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(λ H, natural_transformation.rec_on H (λ η nat, refl (natural_transformation.mk η nat)))
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(λ S, sigma.rec_on S (λ η nat, refl (sigma.mk η nat)))))-/
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sorry
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protected definition id_right (η : F ⟹ G) [fext : funext.{l_1 l_4}] :
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η ∘n id = η :=
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--Proof broken like all trunc_pi proofs
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/-begin
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apply (rec_on η), intros (f, H),
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fapply (path.dcongr_arg2 mk),
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apply (funext.path_forall _ f (λa, !id_right)),
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assert (H1 : is_hprop (Π {a b : C} (g : hom a b), G g ∘ f a = f b ∘ F g)),
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repeat (apply trunc_pi; intros),
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apply (succ_is_trunc -1 (G a_2 ∘ f a) (f a_1 ∘ F a_2)),
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apply (!is_hprop.elim),
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end-/
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sorry
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protected definition to_hset [fx : funext] : is_hset (F ⟹ G) :=
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--Proof broken like all trunc_pi proofs
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/-begin
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protected definition to_hset : is_hset (F ⟹ G) :=
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begin
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apply trunc_equiv, apply (equiv.to_is_equiv sigma_char),
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apply trunc_sigma,
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apply trunc_pi, intro a, exact (@homH (objects D) _ (F a) (G a)),
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intro η, apply trunc_pi, intro a,
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apply trunc_pi, intro b, apply trunc_pi, intro f,
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apply succ_is_trunc, apply trunc_succ, exact (@homH (objects D) _ (F a) (G b)),
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end-/
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sorry
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end
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end natural_transformation
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