160 lines
5.7 KiB
Text
160 lines
5.7 KiB
Text
-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
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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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open function precategory eq prod equiv is_equiv sigma sigma.ops
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structure functor (C D : Precategory) : Type :=
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(obF : C → D)
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(homF : Π ⦃a b : C⦄, hom a b → hom (obF a) (obF b))
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(respect_id : Π (a : C), homF (ID a) = ID (obF a))
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(respect_comp : Π {a b c : C} (g : hom b c) (f : hom a b),
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homF (g ∘ f) = homF g ∘ homF f)
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infixl `⇒`:25 := functor
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namespace functor
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variables {C D E : Precategory}
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coercion [persistent] obF
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coercion [persistent] homF
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-- "functor C D" is equivalent to a certain sigma type
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set_option unifier.max_steps 38500
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protected definition sigma_char :
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(Σ (obF : C → D)
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(homF : Π ⦃a b : C⦄, hom a b → hom (obF a) (obF b)),
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(Π (a : C), homF (ID a) = ID (obF a)) ×
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(Π {a b c : C} (g : hom b c) (f : hom a b),
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homF (g ∘ f) = homF g ∘ homF f)) ≃ (functor C D) :=
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begin
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fapply equiv.mk,
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intro S, fapply functor.mk,
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exact (S.1), exact (S.2.1),
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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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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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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 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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(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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infixr `∘f`:60 := compose
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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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/-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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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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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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mk (λ a b, functor a b)
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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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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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end ops
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end category-/
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namespace functor
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-- open category.ops
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-- universes l m
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variables {C D : Precategory}
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-- check hom C D
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-- variables (F : C ⟶ D) (G : D ⇒ C)
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-- check G ∘ F
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-- check F ∘f G
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-- variables (a b : C) (f : a ⟶ b)
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-- check F a
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-- check F b
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-- check F f
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-- check G (F f)
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-- print "---"
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-- -- check (G ∘ F) f --error
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-- check (λ(x : functor C C), x) (G ∘ F) f
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-- check (G ∘f F) f
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-- print "---"
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-- -- check (G ∘ F) a --error
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-- check (G ∘f F) a
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-- print "---"
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-- -- check λ(H : hom C D) (x : C), H x --error
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-- check λ(H : @hom _ Cat C D) (x : C), H x
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-- check λ(H : C ⇒ D) (x : C), H x
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-- print "---"
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-- -- variables {obF obG : C → D} (Hob : ∀x, obF x = obG x) (homF : Π(a b : C) (f : a ⟶ b), obF a ⟶ obF b) (homG : Π(a b : C) (f : a ⟶ b), obG a ⟶ obG b)
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-- -- check eq.rec_on (funext Hob) homF = homG
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/-theorem mk_heq {obF obG : C → D} {homF homG idF idG compF compG} (Hob : ∀x, obF x = obG x)
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(Hmor : ∀(a b : C) (f : a ⟶ b), homF a b f == homG a b f)
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: mk obF homF idF compF = mk obG homG idG compG :=
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hddcongr_arg4 mk
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(funext Hob)
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(hfunext (λ a, hfunext (λ b, hfunext (λ f, !Hmor))))
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!proof_irrel
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!proof_irrel
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protected theorem hequal {F G : C ⇒ D} : Π (Hob : ∀x, F x = G x)
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(Hmor : ∀a b (f : a ⟶ b), F f == G f), F = G :=
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functor.rec
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(λ obF homF idF compF,
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functor.rec
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(λ obG homG idG compG Hob Hmor, mk_heq Hob Hmor)
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G)
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F-/
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-- theorem mk_eq {obF obG : C → D} {homF homG idF idG compF compG} (Hob : ∀x, obF x = obG x)
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-- (Hmor : ∀(a b : C) (f : a ⟶ b), cast (congr_arg (λ x, x a ⟶ x b) (funext Hob)) (homF a b f)
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-- = homG a b f)
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-- : mk obF homF idF compF = mk obG homG idG compG :=
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-- dcongr_arg4 mk
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-- (funext Hob)
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-- (funext (λ a, funext (λ b, funext (λ f, sorry ⬝ Hmor a b f))))
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-- -- to fill this sorry use (a generalization of) cast_pull
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-- !proof_irrel
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-- !proof_irrel
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-- protected theorem equal {F G : C ⇒ D} : Π (Hob : ∀x, F x = G x)
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-- (Hmor : ∀a b (f : a ⟶ b), cast (congr_arg (λ x, x a ⟶ x b) (funext Hob)) (F f) = G f), F = G :=
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-- functor.rec
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-- (λ obF homF idF compF,
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-- functor.rec
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-- (λ obG homG idG compG Hob Hmor, mk_eq Hob Hmor)
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-- G)
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-- F
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end functor
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