lean2/library/algebra/category/functor.lean

-- 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
open function
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