lean2/hott/algebra/precategory/functor.hlean

/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Module: algebra.precategory.functor
Authors: Floris van Doorn, Jakob von Raumer
-/

import .basic types.pi .iso

open function category eq prod equiv is_equiv sigma sigma.ops is_trunc funext iso
open pi




  section

  variables {A : Type} {B : A → Type} {C : Πa, B a → Type} {D : Πa b, C a b → Type}

  definition homotopy2 [reducible] (f g : Πa b, C a b) : Type :=
  Πa b, f a b = g a b

  definition homotopy3 [reducible] (f g : Πa b c, D a b c) : Type :=
  Πa b c, f a b c = g a b c
  notation f `∼2`:50 g := homotopy2 f g
  notation f `∼3`:50 g := homotopy3 f g

  -- definition apD100 {f g : Πa b, C a b} (p : f = g) : f ∼2 g :=
  -- λa b, eq.rec_on p idp

  definition apD100 {f g : Πa b, C a b} (p : f = g) : f ∼2 g :=
  λa b, apD10 (apD10 p a) b

  definition apD1000 {f g : Πa b c, D a b c} (p : f = g) : f ∼3 g :=
  λa b c, apD100 (apD10 p a) b c

  definition eq_of_homotopy2 {f g : Πa b, C a b} (H : f ∼2 g) : f = g :=
  eq_of_homotopy (λa, eq_of_homotopy (H a))

  definition eq_of_homotopy3 {f g : Πa b c, D a b c} (H : f ∼3 g) : f = g :=
  eq_of_homotopy (λa, eq_of_homotopy2 (H a))

  definition eq_of_homotopy2_id (f : Πa b, C a b)
    : eq_of_homotopy2 (λa b, idpath (f a b)) = idpath f :=
  begin
    apply concat,
      {apply (ap (λx, eq_of_homotopy x)), apply eq_of_homotopy, intro a, apply eq_of_homotopy_id},
    apply eq_of_homotopy_id
  end

  definition eq_of_homotopy3_id (f : Πa b c, D a b c)
    : eq_of_homotopy3 (λa b c, idpath (f a b c)) = idpath f :=
  begin
    apply concat,
      {apply (ap (λx, eq_of_homotopy x)), apply eq_of_homotopy, intro a, apply eq_of_homotopy2_id},
    apply eq_of_homotopy_id
  end

  --TODO: put in namespace funext
  definition is_equiv_apD100 [instance] (f g : Πa b, C a b) : is_equiv (@apD100 A B C f g) :=
  adjointify _
             eq_of_homotopy2
             begin
               intro H, esimp {apD100,eq_of_homotopy2, function.compose},
               apply eq_of_homotopy, intro a,
               apply concat, apply (ap (λx, @apD10 _ (λb : B a, _) _ _ (x a))), apply (retr apD10),
--TODO: remove implicit argument after #469 is closed
               apply (retr apD10)
             end
             begin
               intro p, cases p, apply eq_of_homotopy2_id
             end

  definition is_equiv_apD1000 [instance] (f g : Πa b c, D a b c) : is_equiv (@apD1000 A B C D f g) :=
  adjointify _
             eq_of_homotopy3
             begin
               intro H, apply eq_of_homotopy, intro a,
               apply concat, {apply (ap (λx, @apD100 _ _ (λ(b : B a)(c : C a b), _) _ _ (x a))), apply (retr apD10)},
--TODO: remove implicit argument after #469 is closed
               apply (@retr _ _ apD100 !is_equiv_apD100) --is explicit argument needed here?
             end
             begin
               intro p, cases p, apply eq_of_homotopy3_id
             end

  protected definition homotopy2.rec_on {f g : Πa b, C a b} {P : (f ∼2 g) → Type}
    (p : f ∼2 g) (H : Π(q : f = g), P (apD100 q)) : P p :=
  retr apD100 p ▹ H (eq_of_homotopy2 p)

  protected definition homotopy3.rec_on {f g : Πa b c, D a b c} {P : (f ∼3 g) → Type}
    (p : f ∼3 g) (H : Π(q : f = g), P (apD1000 q)) : P p :=
  retr apD1000 p ▹ H (eq_of_homotopy3 p)
  end



structure functor (C D : Precategory) : Type :=
  (to_fun_ob : C → D)
  (to_fun_hom : Π ⦃a b : C⦄, hom a b → hom (to_fun_ob a) (to_fun_ob b))
  (respect_id : Π (a : C), to_fun_hom (ID a) = ID (to_fun_ob a))
  (respect_comp : Π {a b c : C} (g : hom b c) (f : hom a b),
    to_fun_hom (g ∘ f) = to_fun_hom g ∘ to_fun_hom f)

namespace functor

  infixl `⇒`:25 := functor
  variables {C D E : Precategory}

  attribute to_fun_ob [coercion]
  attribute to_fun_hom [coercion]

  -- The following lemmas will later be used to prove that the type of
  -- precategories forms a precategory itself
  protected definition compose [reducible] (G : functor D E) (F : functor C D) : functor C E :=
  functor.mk
    (λ x, G (F x))
    (λ a b f, G (F f))
    (λ a, calc
      G (F (ID a)) = G (ID (F a)) : by rewrite respect_id
               ... = ID (G (F a)) : by rewrite respect_id)
    (λ a b c g f, calc
      G (F (g ∘ f)) = G (F g ∘ F f)     : by rewrite respect_comp
                ... = G (F g) ∘ G (F f) : by rewrite respect_comp)

  infixr `∘f`:60 := compose

  protected definition id [reducible] {C : Precategory} : functor C C :=
  mk (λa, a) (λ a b f, f) (λ a, idp) (λ a b c f g, idp)

  protected definition ID [reducible] (C : Precategory) : functor C C := id

  definition functor_eq_mk'' {F₁ F₂ : C → D} {H₁ : Π(a b : C), hom a b → hom (F₁ a) (F₁ b)}
    {H₂ : Π(a b : C), hom a b → hom (F₂ a) (F₂ b)} (id₁ id₂ comp₁ comp₂)
    (pF : F₁ = F₂) (pH : pF ▹ H₁ = H₂)
      : functor.mk F₁ H₁ id₁ comp₁ = functor.mk F₂ H₂ id₂ comp₂ :=
  apD01111 functor.mk pF pH !is_hprop.elim !is_hprop.elim

  definition functor_eq_mk' {F₁ F₂ : C → D} {H₁ : Π(a b : C), hom a b → hom (F₁ a) (F₁ b)}
    {H₂ : Π(a b : C), hom a b → hom (F₂ a) (F₂ b)} (id₁ id₂ comp₁ comp₂) (pF : F₁ ∼ F₂)
    (pH : Π(a b : C) (f : hom a b), hom_of_eq (pF b) ∘ H₁ a b f ∘ inv_of_eq (pF a) = H₂ a b f)
      : functor.mk F₁ H₁ id₁ comp₁ = functor.mk F₂ H₂ id₂ comp₂ :=
  functor_eq_mk'' id₁ id₂ comp₁ comp₂ (eq_of_homotopy pF)
    (eq_of_homotopy (λc, eq_of_homotopy (λc', eq_of_homotopy (λf,
      begin
        apply concat, rotate_left 1, exact (pH c c' f),
        apply concat, rotate_left 1, apply transport_hom,
        apply concat, rotate_left 1,
        exact (pi_transport_constant (eq_of_homotopy pF) (H₁ c c') f),
        apply (apD10' f),
        apply concat, rotate_left 1,
        exact (pi_transport_constant (eq_of_homotopy pF) (H₁ c) c'),
        apply (apD10' c'),
        apply concat, rotate_left 1,
        exact (pi_transport_constant (eq_of_homotopy pF) H₁ c),
        apply idp
      end))))

  definition functor_eq_mk_constant {F : C → D} {H₁ : Π(a b : C), hom a b → hom (F a) (F b)}
    {H₂ : Π(a b : C), hom a b → hom (F a) (F b)} (id₁ id₂ comp₁ comp₂)
    (pH : Π(a b : C) (f : hom a b), H₁ a b f = H₂ a b f)
      : functor.mk F H₁ id₁ comp₁ = functor.mk F H₂ id₂ comp₂ :=
  functor_eq_mk'' id₁ id₂ comp₁ comp₂ idp
                  (eq_of_homotopy (λc, eq_of_homotopy (λc', eq_of_homotopy (pH c c'))))

  definition functor_eq_mk {F₁ F₂ : C ⇒ D} : Π(p : to_fun_ob F₁ ∼ to_fun_ob F₂),
    (Π(a b : C) (f : hom a b), hom_of_eq (p b) ∘ F₁ f ∘ inv_of_eq (p a) = F₂ f) → F₁ = F₂ :=
  functor.rec_on F₁ (λO₁ H₁ id₁ comp₁, functor.rec_on F₂ (λO₂ H₂ id₂ comp₂ p, !functor_eq_mk'))

  protected definition assoc {A B C D : Precategory} (H : functor C D) (G : functor B C) (F : functor A B) :
      H ∘f (G ∘f F) = (H ∘f G) ∘f F :=
  !functor_eq_mk_constant (λa b f, idp)

  protected definition id_left  (F : functor C D) : id ∘f F = F :=
  functor.rec_on F (λF1 F2 F3 F4, !functor_eq_mk_constant (λa b f, idp))

  protected definition id_right (F : functor C D) : F ∘f id = F :=
  functor.rec_on F (λF1 F2 F3 F4, !functor_eq_mk_constant (λa b f, idp))

  set_option apply.class_instance false
  -- "functor C D" is equivalent to a certain sigma type
  set_option unifier.max_steps 38500
  protected definition sigma_char :
    (Σ (to_fun_ob : C → D)
    (to_fun_hom : Π ⦃a b : C⦄, hom a b → hom (to_fun_ob a) (to_fun_ob b)),
    (Π (a : C), to_fun_hom (ID a) = ID (to_fun_ob a)) ×
    (Π {a b c : C} (g : hom b c) (f : hom a b),
      to_fun_hom (g ∘ f) = to_fun_hom g ∘ to_fun_hom f)) ≃ (functor C D) :=
  begin
    fapply equiv.MK,
      {intro S, fapply functor.mk,
        exact (S.1), exact (S.2.1),
        exact (pr₁ S.2.2), exact (pr₂ S.2.2)},
      {intro F,
        cases F with (d1, d2, d3, d4),
        exact (sigma.mk d1 (sigma.mk d2 (pair d3 (@d4))))},
      {intro F,
        cases F,
        apply idp},
      {intro S,
        cases S with (d1, S2),
        cases S2 with (d2, P1),
        cases P1,
        apply idp},
  end

  protected definition is_hset_functor
    [HD : is_hset D] : is_hset (functor C D) :=
  begin
    apply is_trunc_is_equiv_closed, apply equiv.to_is_equiv,
      apply sigma_char,
    apply is_trunc_sigma, apply is_trunc_pi, intros, exact HD, intro F,
    apply is_trunc_sigma, apply is_trunc_pi, intro a,
      {apply is_trunc_pi, intro b,
       apply is_trunc_pi, intro c, apply !homH},
    intro H, apply is_trunc_prod,
      {apply is_trunc_pi, intro a,
       apply is_trunc_eq, apply is_trunc_succ, apply !homH},
      {repeat (apply is_trunc_pi; intros),
       apply is_trunc_eq, apply is_trunc_succ, apply !homH},
  end

  --set_option pp.universes true
  -- set_option pp.notation false
  -- set_option pp.implicit true
  definition functor_eq2' {obF obF' : C → D} {homF homF' idF idF' compF compF'}
    (p q : functor.mk obF homF idF compF = functor.mk obF' homF' idF' compF') (r : obF = obF')
    : p = q :=
  begin
    cases r,
  end

  definition functor_eq2 {F₁ F₂ : C ⇒ D} (p q : F₁ = F₂) (r : ap010 to_fun_ob p ∼ ap010 to_fun_ob q)
    : p = q :=
  begin

  end

  -- definition ap010_functor_eq_mk' {F₁ F₂ : C ⇒ D} (p : to_fun_ob F₁ = to_fun_ob F₂)
  --   (q : p ▹ F₁ = F₂) (c : C) :
  --   ap to_fun_ob (functor_eq_mk (apD10 p) (λa b f, _)) = p := sorry
  -- begin
  --   cases F₂, revert q, apply (homotopy.rec_on p), clear p, esimp, intros (p, q),
  --   cases p, clears (e_1, e_2),
  -- end

  -- TODO: remove sorry
  -- maybe some lemma "recursion on homotopy (and equiv)" could be useful
  definition ap010_functor_eq_mk {F₁ F₂ : C ⇒ D} (p : to_fun_ob F₁ ∼ to_fun_ob F₂)
    (q : (λ(a b : C) (f : hom a b), hom_of_eq (p b) ∘ F₁ f ∘ inv_of_eq (p a)) ∼3 to_fun_hom F₂) (c : C) :
    ap010 to_fun_ob (functor_eq_mk p q) c = p c :=
  begin
    cases F₂, revert q, apply (homotopy.rec_on p), clear p, esimp, intros (p, q),
    apply sorry,
    --cases p, clears (e_1, e_2, p),

    --exact (homotopy3.rec_on q sorry)
--    apply (homotopy3.rec_on q),
  end

  -- definition ap010_functor_eq_mk {F₁ F₂ : C ⇒ D} (p : to_fun_ob F₁ ∼ to_fun_ob F₂)
  --   (q : Π(a b : C) (f : hom a b), hom_of_eq (p b) ∘ F₁ f ∘ inv_of_eq (p a) = F₂ f) (c : C) :
  --   ap010 to_fun_ob (functor_eq_mk p q) c = p c :=
  -- begin
  --   cases F₂, revert q, apply (homotopy.rec_on p), clear p, esimp, intros (p, q),
  --   cases p, clears (e_1, e_2, p),
  --   apply (homotopy3.rec_on q),
  -- end
-- ⊢ ap010 to_fun_ob (functor_eq_mk rfl q) c = rfl

end functor

namespace category
  open functor

  --TODO: make this a structure
  definition precat_strict_precat : precategory (Σ (C : Precategory), is_hset C) :=
  precategory.mk (λ a b, functor a.1 b.1)
     (λ a b, @functor.is_hset_functor a.1 b.1 b.2)
     (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 Precat_of_strict_cats := precategory.Mk precat_strict_precat

  namespace ops

    abbreviation SPreCat := Precat_of_strict_cats
    --attribute precat_strict_precat [instance]

  end ops

end category