lean2/hott/algebra/precategory/nat_trans.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.nat_trans
Author: Floris van Doorn, Jakob von Raumer
-/

import .functor .morphism
open eq category functor is_trunc equiv sigma.ops sigma is_equiv function pi funext

structure nat_trans {C D : Precategory} (F G : C ⇒ D) :=
 (natural_map : Π (a : C), hom (F a) (G a))
 (naturality : Π {a b : C} (f : hom a b), G f ∘ natural_map a = natural_map b ∘ F f)

namespace nat_trans

  infixl `⟹`:25 := nat_trans -- \==>
  variables {C D : Precategory} {F G H I : C ⇒ D}

  attribute natural_map [coercion]

  protected definition compose [reducible] (η : G ⟹ H) (θ : F ⟹ G) : F ⟹ H :=
  nat_trans.mk
    (λ a, η a ∘ θ a)
    (λ a b f,
      calc
        H f ∘ (η a ∘ θ a) = (H f ∘ η a) ∘ θ a : assoc
                      ... = (η b ∘ G f) ∘ θ a : naturality η f
                      ... = η b ∘ (G f ∘ θ a) : assoc
                      ... = η b ∘ (θ b ∘ F f) : naturality θ f
                      ... = (η b ∘ θ b) ∘ F f : assoc)

  infixr `∘n`:60 := compose

  protected definition id [reducible] {C D : Precategory} {F : functor C D} : nat_trans F F :=
  mk (λa, id) (λa b f, !id_right ⬝ !id_left⁻¹)

  protected definition ID [reducible] {C D : Precategory} (F : functor C D) : nat_trans F F :=
  id

  local attribute is_hprop_eq_hom [instance]
  definition nat_trans_eq_mk' {η₁ η₂ : Π (a : C), hom (F a) (G a)}
    (nat₁ : Π (a b : C) (f : hom a b), G f ∘ η₁ a = η₁ b ∘ F f)
    (nat₂ : Π (a b : C) (f : hom a b), G f ∘ η₂ a = η₂ b ∘ F f)
    (p : η₁ ∼ η₂)
      : nat_trans.mk η₁ nat₁ = nat_trans.mk η₂ nat₂ :=
  apD011 nat_trans.mk (eq_of_homotopy p) !is_hprop.elim

  definition nat_trans_eq_mk {η₁ η₂ : F ⟹ G} : natural_map η₁ ∼ natural_map η₂ → η₁ = η₂ :=
  nat_trans.rec_on η₁ (λf₁ nat₁, nat_trans.rec_on η₂ (λf₂ nat₂ p, !nat_trans_eq_mk' p))

  protected definition assoc (η₃ : H ⟹ I) (η₂ : G ⟹ H) (η₁ : F ⟹ G) :
      η₃ ∘n (η₂ ∘n η₁) = (η₃ ∘n η₂) ∘n η₁ :=
  nat_trans_eq_mk (λa, !assoc)

  protected definition id_left (η : F ⟹ G) : id ∘n η = η :=
  nat_trans_eq_mk (λa, !id_left)

  protected definition id_right (η : F ⟹ G) : η ∘n id = η :=
  nat_trans_eq_mk (λa, !id_right)

  protected definition sigma_char (F G : C ⇒ D) :
    (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) :=
  begin
    fapply equiv.mk,
      intro S, apply nat_trans.mk, exact (S.2),
    fapply adjointify,
      intro H,
          fapply sigma.mk,
            intro a, exact (H a),
          intros (a, b, f), exact (naturality H f),
    intro η, apply nat_trans_eq_mk, intro a, apply idp,
    intro S,
    fapply sigma_eq,
      apply eq_of_homotopy, intro a,
      apply idp,
    apply is_hprop.elim,
  end

  set_option apply.class_instance false
  definition is_hset_nat_trans : is_hset (F ⟹ G) :=
  begin
    apply is_trunc_is_equiv_closed, apply (equiv.to_is_equiv !sigma_char),
    apply is_trunc_sigma,
      apply is_trunc_pi, intro a, exact (@homH (Precategory.carrier D) _ (F a) (G a)),
    intro η, apply is_trunc_pi, intro a,
    apply is_trunc_pi, intro b, apply is_trunc_pi, intro f,
    apply is_trunc_eq, apply is_trunc_succ, exact (@homH (Precategory.carrier D) _ (F a) (G b)),
  end

end nat_trans