91 lines
3.5 KiB
Text
91 lines
3.5 KiB
Text
/-
|
||
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 .iso
|
||
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 : by rewrite assoc
|
||
... = (η b ∘ G f) ∘ θ a : by rewrite naturality
|
||
... = η b ∘ (G f ∘ θ a) : by rewrite assoc
|
||
... = η b ∘ (θ b ∘ F f) : by rewrite naturality
|
||
... = (η b ∘ θ b) ∘ F f : by rewrite 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
|
||
|
||
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
|