lean2/library/algebra/category/functor.lean

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-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Floris van Doorn
import .basic
import logic.cast
import algebra.function --remove if "typeof" is moved
open function --remove if "typeof" is moved
open category eq eq.ops heq
inductive functor (C D : Category) : Type :=
mk : Π (obF : C → D) (homF : Π(a b : C), hom a b → hom (obF a) (obF b)),
(Π (a : C), homF a a (ID a) = ID (obF a)) →
(Π (a b c : C) {g : hom b c} {f : hom a b}, homF a c (g ∘ f) = homF b c g ∘ homF a b f) →
functor C D
infixl `⇒`:25 := functor
namespace functor
variables {C D E : Category}
definition object [coercion] (F : functor C D) : C → D := rec (λ obF homF Hid Hcomp, obF) F
definition morphism [coercion] (F : functor C D) : Π⦃a b : C⦄, hom a b → hom (F a) (F b) :=
rec (λ obF homF Hid Hcomp, homF) F
theorem respect_id (F : functor C D) : Π (a : C), F (ID a) = id :=
rec (λ obF homF Hid Hcomp, Hid) F
theorem respect_comp (F : functor C D) : Π ⦃a b c : C⦄ (g : hom b c) (f : hom a b),
F (g ∘ f) = F g ∘ F f :=
rec (λ obF homF Hid Hcomp, Hcomp) F
protected definition compose (G : functor D E) (F : functor C D) : functor C E :=
functor.mk
(λx, G (F x))
(λ a b f, G (F f))
(λ a, calc
G (F (ID a)) = G id : {respect_id F a} --not giving the braces explicitly makes the elaborator compute a couple more seconds
... = id : respect_id G (F a))
(λ a b c g f, calc
G (F (g ∘ f)) = G (F g ∘ F f) : respect_comp F g f
... = G (F g) ∘ G (F f) : respect_comp G (F g) (F f))
infixr `∘f`:60 := compose
protected theorem assoc {A B C D : Category} (H : functor C D) (G : functor B C) (F : functor A B) :
H ∘f (G ∘f F) = (H ∘f G) ∘f F :=
rfl
protected definition id {C : Category} : functor C C :=
mk (λa, a) (λ a b f, f) (λ a, rfl) (λ a b c f g, rfl)
protected definition ID (C : Category) : functor C C := id
protected theorem id_left (F : functor C D) : id ∘f F = F :=
functor.rec (λ obF homF idF compF, dcongr_arg4 mk rfl rfl !proof_irrel !proof_irrel) F
protected theorem id_right (F : functor C D) : F ∘f id = F :=
functor.rec (λ obF homF idF compF, dcongr_arg4 mk rfl rfl !proof_irrel !proof_irrel) F
end functor
namespace category
open functor
definition category_of_categories [reducible] : category Category :=
mk (λ a b, functor a b)
(λ a b c g f, functor.compose g f)
(λ a, functor.id)
(λ a b c d h g f, !functor.assoc)
(λ a b f, !functor.id_left)
(λ a b f, !functor.id_right)
definition Category_of_categories [reducible] := Mk category_of_categories
namespace ops
notation `Cat`:max := Category_of_categories
instance [persistent] category_of_categories
end ops
end category
namespace functor
-- open category.ops
-- universes l m
variables {C D : Category}
-- check hom C D
-- variables (F : C ⟶ D) (G : D ⇒ C)
-- check G ∘ F
-- check F ∘f G
-- variables (a b : C) (f : a ⟶ b)
-- check F a
-- check F b
-- check F f
-- check G (F f)
-- print "---"
-- -- check (G ∘ F) f --error
-- check (λ(x : functor C C), x) (G ∘ F) f
-- check (G ∘f F) f
-- print "---"
-- -- check (G ∘ F) a --error
-- check (G ∘f F) a
-- print "---"
-- -- check λ(H : hom C D) (x : C), H x --error
-- check λ(H : @hom _ Cat C D) (x : C), H x
-- check λ(H : C ⇒ D) (x : C), H x
-- print "---"
-- -- 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)
-- -- check eq.rec_on (funext Hob) homF = homG
theorem mk_heq {obF obG : C → D} {homF homG idF idG compF compG} (Hob : ∀x, obF x = obG x)
(Hmor : ∀(a b : C) (f : a ⟶ b), homF a b f == homG a b f)
: mk obF homF idF compF = mk obG homG idG compG :=
hddcongr_arg4 mk
(funext Hob)
(hfunext (λ a, hfunext (λ b, hfunext (λ f, !Hmor))))
!proof_irrel
!proof_irrel
protected theorem hequal {F G : C ⇒ D} : Π (Hob : ∀x, F x = G x)
(Hmor : ∀a b (f : a ⟶ b), F f == G f), F = G :=
functor.rec
(λ obF homF idF compF,
functor.rec
(λ obG homG idG compG Hob Hmor, mk_heq Hob Hmor)
G)
F
-- theorem mk_eq {obF obG : C → D} {homF homG idF idG compF compG} (Hob : ∀x, obF x = obG x)
-- (Hmor : ∀(a b : C) (f : a ⟶ b), cast (congr_arg (λ x, x a ⟶ x b) (funext Hob)) (homF a b f)
-- = homG a b f)
-- : mk obF homF idF compF = mk obG homG idG compG :=
-- dcongr_arg4 mk
-- (funext Hob)
-- (funext (λ a, funext (λ b, funext (λ f, sorry ⬝ Hmor a b f))))
-- -- to fill this sorry use (a generalization of) cast_pull
-- !proof_irrel
-- !proof_irrel
-- protected theorem equal {F G : C ⇒ D} : Π (Hob : ∀x, F x = G x)
-- (Hmor : ∀a b (f : a ⟶ b), cast (congr_arg (λ x, x a ⟶ x b) (funext Hob)) (F f) = G f), F = G :=
-- functor.rec
-- (λ obF homF idF compF,
-- functor.rec
-- (λ obG homG idG compG Hob Hmor, mk_eq Hob Hmor)
-- G)
-- F
end functor