206 lines
9.1 KiB
Text
206 lines
9.1 KiB
Text
/-
|
||
Copyright (c) 2015 Floris van Doorn. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Authors: Floris van Doorn
|
||
-/
|
||
|
||
--note: modify definition in category.set
|
||
import .constructions.functor .constructions.hset .constructions.product .constructions.opposite
|
||
|
||
open category eq category.ops functor prod.ops is_trunc iso
|
||
|
||
namespace yoneda
|
||
set_option class.conservative false
|
||
|
||
definition representable_functor_assoc [C : Precategory] {a1 a2 a3 a4 a5 a6 : C}
|
||
(f1 : hom a5 a6) (f2 : hom a4 a5) (f3 : hom a3 a4) (f4 : hom a2 a3) (f5 : hom a1 a2)
|
||
: (f1 ∘ f2) ∘ f3 ∘ (f4 ∘ f5) = f1 ∘ (f2 ∘ f3 ∘ f4) ∘ f5 :=
|
||
calc
|
||
_ = f1 ∘ f2 ∘ f3 ∘ f4 ∘ f5 : by rewrite -assoc
|
||
... = f1 ∘ (f2 ∘ f3) ∘ f4 ∘ f5 : by rewrite -assoc
|
||
... = f1 ∘ ((f2 ∘ f3) ∘ f4) ∘ f5 : by rewrite -(assoc (f2 ∘ f3) _ _)
|
||
... = _ : by rewrite (assoc f2 f3 f4)
|
||
|
||
definition hom_functor (C : Precategory) : Cᵒᵖ ×c C ⇒ set :=
|
||
functor.mk (λ(x : Cᵒᵖ ×c C), homset x.1 x.2)
|
||
(λ(x y : Cᵒᵖ ×c C) (f : _) (h : homset x.1 x.2), f.2 ∘⁅ C ⁆ (h ∘⁅ C ⁆ f.1))
|
||
begin
|
||
intro x, apply eq_of_homotopy, intro h, exact (!id_left ⬝ !id_right)
|
||
end
|
||
begin
|
||
intro x y z g f, apply eq_of_homotopy, intro h,
|
||
exact (representable_functor_assoc g.2 f.2 h f.1 g.1),
|
||
end
|
||
end yoneda
|
||
|
||
|
||
open is_equiv equiv
|
||
|
||
namespace functor
|
||
open prod nat_trans
|
||
variables {C D E : Precategory} (F : C ×c D ⇒ E) (G : C ⇒ E ^c D)
|
||
definition functor_curry_ob [reducible] (c : C) : E ^c D :=
|
||
functor.mk (λd, F (c,d))
|
||
(λd d' g, F (id, g))
|
||
(λd, !respect_id)
|
||
(λd₁ d₂ d₃ g' g, calc
|
||
F (id, g' ∘ g) = F (id ∘ id, g' ∘ g) : by rewrite id_comp
|
||
... = F ((id,g') ∘ (id, g)) : by esimp
|
||
... = F (id,g') ∘ F (id, g) : by rewrite respect_comp)
|
||
|
||
local abbreviation Fob := @functor_curry_ob
|
||
|
||
definition functor_curry_hom ⦃c c' : C⦄ (f : c ⟶ c') : Fob F c ⟹ Fob F c' :=
|
||
nat_trans.mk (λd, F (f, id))
|
||
(λd d' g, calc
|
||
F (id, g) ∘ F (f, id) = F (id ∘ f, g ∘ id) : respect_comp F
|
||
... = F (f, g ∘ id) : by rewrite id_left
|
||
... = F (f, g) : by rewrite id_right
|
||
... = F (f ∘ id, g) : by rewrite id_right
|
||
... = F (f ∘ id, id ∘ g) : by rewrite id_left
|
||
... = F (f, id) ∘ F (id, g) : (respect_comp F (f, id) (id, g))⁻¹ᵖ)
|
||
|
||
local abbreviation Fhom := @functor_curry_hom
|
||
|
||
theorem functor_curry_hom_def ⦃c c' : C⦄ (f : c ⟶ c') (d : D) :
|
||
(Fhom F f) d = to_fun_hom F (f, id) := idp
|
||
|
||
theorem functor_curry_id (c : C) : Fhom F (ID c) = nat_trans.id :=
|
||
nat_trans_eq (λd, respect_id F _)
|
||
|
||
theorem functor_curry_comp ⦃c c' c'' : C⦄ (f' : c' ⟶ c'') (f : c ⟶ c')
|
||
: Fhom F (f' ∘ f) = Fhom F f' ∘n Fhom F f :=
|
||
nat_trans_eq (λd, calc
|
||
natural_map (Fhom F (f' ∘ f)) d = F (f' ∘ f, id) : by rewrite functor_curry_hom_def
|
||
... = F (f' ∘ f, id ∘ id) : by rewrite id_comp
|
||
... = F ((f',id) ∘ (f, id)) : by esimp
|
||
... = F (f',id) ∘ F (f, id) : by rewrite [respect_comp F]
|
||
... = natural_map ((Fhom F f') ∘ (Fhom F f)) d : by esimp)
|
||
|
||
definition functor_curry [reducible] : C ⇒ E ^c D :=
|
||
functor.mk (functor_curry_ob F)
|
||
(functor_curry_hom F)
|
||
(functor_curry_id F)
|
||
(functor_curry_comp F)
|
||
|
||
definition functor_uncurry_ob [reducible] (p : C ×c D) : E :=
|
||
to_fun_ob (G p.1) p.2
|
||
local abbreviation Gob := @functor_uncurry_ob
|
||
|
||
definition functor_uncurry_hom ⦃p p' : C ×c D⦄ (f : hom p p') : Gob G p ⟶ Gob G p' :=
|
||
to_fun_hom (to_fun_ob G p'.1) f.2 ∘ natural_map (to_fun_hom G f.1) p.2
|
||
local abbreviation Ghom := @functor_uncurry_hom
|
||
|
||
theorem functor_uncurry_id (p : C ×c D) : Ghom G (ID p) = id :=
|
||
calc
|
||
Ghom G (ID p) = to_fun_hom (to_fun_ob G p.1) id ∘ natural_map (to_fun_hom G id) p.2 : by esimp
|
||
... = id ∘ natural_map (to_fun_hom G id) p.2 : by rewrite respect_id
|
||
... = id ∘ natural_map nat_trans.id p.2 : by rewrite respect_id
|
||
... = id : id_comp
|
||
|
||
theorem functor_uncurry_comp ⦃p p' p'' : C ×c D⦄ (f' : p' ⟶ p'') (f : p ⟶ p')
|
||
: Ghom G (f' ∘ f) = Ghom G f' ∘ Ghom G f :=
|
||
calc
|
||
Ghom G (f' ∘ f)
|
||
= to_fun_hom (to_fun_ob G p''.1) (f'.2 ∘ f.2) ∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : by esimp
|
||
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
|
||
∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : by rewrite respect_comp
|
||
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
|
||
∘ natural_map (to_fun_hom G f'.1 ∘ to_fun_hom G f.1) p.2 : by rewrite respect_comp
|
||
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
|
||
∘ (natural_map (to_fun_hom G f'.1) p.2 ∘ natural_map (to_fun_hom G f.1) p.2) : by esimp
|
||
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
|
||
∘ (natural_map (to_fun_hom G f'.1) p.2 ∘ natural_map (to_fun_hom G f.1) p.2) : by esimp
|
||
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ natural_map (to_fun_hom G f'.1) p'.2)
|
||
∘ (to_fun_hom (to_fun_ob G p'.1) f.2 ∘ natural_map (to_fun_hom G f.1) p.2) :
|
||
by rewrite [square_prepostcompose (!naturality⁻¹ᵖ) _ _]
|
||
... = Ghom G f' ∘ Ghom G f : by esimp
|
||
|
||
definition functor_uncurry [reducible] : C ×c D ⇒ E :=
|
||
functor.mk (functor_uncurry_ob G)
|
||
(functor_uncurry_hom G)
|
||
(functor_uncurry_id G)
|
||
(functor_uncurry_comp G)
|
||
|
||
theorem functor_uncurry_functor_curry : functor_uncurry (functor_curry F) = F :=
|
||
functor_eq (λp, ap (to_fun_ob F) !prod.eta)
|
||
begin
|
||
intro cd cd' fg,
|
||
cases cd with c d, cases cd' with c' d', cases fg with f g,
|
||
transitivity to_fun_hom (functor_uncurry (functor_curry F)) (f, g),
|
||
apply id_leftright,
|
||
show (functor_uncurry (functor_curry F)) (f, g) = F (f,g),
|
||
from calc
|
||
(functor_uncurry (functor_curry F)) (f, g) = to_fun_hom F (id, g) ∘ to_fun_hom F (f, id) : by esimp
|
||
... = F (id ∘ f, g ∘ id) : by krewrite [respect_comp F (id,g) (f,id)]
|
||
... = F (f, g ∘ id) : by rewrite id_left
|
||
... = F (f,g) : by rewrite id_right,
|
||
end
|
||
|
||
definition functor_curry_functor_uncurry_ob (c : C)
|
||
: functor_curry (functor_uncurry G) c = G c :=
|
||
begin
|
||
fapply functor_eq,
|
||
{intro d, reflexivity},
|
||
{intro d d' g,
|
||
apply concat, apply id_leftright,
|
||
show to_fun_hom (functor_curry (functor_uncurry G) c) g = to_fun_hom (G c) g,
|
||
from calc
|
||
to_fun_hom (functor_curry (functor_uncurry G) c) g
|
||
= to_fun_hom (G c) g ∘ natural_map (to_fun_hom G (ID c)) d : by esimp
|
||
... = to_fun_hom (G c) g ∘ natural_map (ID (G c)) d : by rewrite respect_id
|
||
... = to_fun_hom (G c) g ∘ id : by reflexivity
|
||
... = to_fun_hom (G c) g : by rewrite id_right}
|
||
end
|
||
|
||
theorem functor_curry_functor_uncurry : functor_curry (functor_uncurry G) = G :=
|
||
begin
|
||
fapply functor_eq, exact (functor_curry_functor_uncurry_ob G),
|
||
intro c c' f,
|
||
fapply nat_trans_eq,
|
||
intro d,
|
||
apply concat,
|
||
{apply (ap (λx, x ∘ _)),
|
||
apply concat, apply natural_map_hom_of_eq, apply (ap hom_of_eq), apply ap010_functor_eq},
|
||
apply concat,
|
||
{apply (ap (λx, _ ∘ x)), apply (ap (λx, _ ∘ x)),
|
||
apply concat, apply natural_map_inv_of_eq,
|
||
apply (ap (λx, hom_of_eq x⁻¹)), apply ap010_functor_eq},
|
||
apply concat, apply id_leftright,
|
||
apply concat, apply (ap (λx, x ∘ _)), apply respect_id,
|
||
apply id_left
|
||
end
|
||
|
||
definition prod_functor_equiv_functor_functor (C D E : Precategory)
|
||
: (C ×c D ⇒ E) ≃ (C ⇒ E ^c D) :=
|
||
equiv.MK functor_curry
|
||
functor_uncurry
|
||
functor_curry_functor_uncurry
|
||
functor_uncurry_functor_curry
|
||
|
||
|
||
definition functor_prod_flip (C D : Precategory) : C ×c D ⇒ D ×c C :=
|
||
functor.mk (λp, (p.2, p.1))
|
||
(λp p' h, (h.2, h.1))
|
||
(λp, idp)
|
||
(λp p' p'' h' h, idp)
|
||
|
||
definition functor_prod_flip_functor_prod_flip (C D : Precategory)
|
||
: functor_prod_flip D C ∘f (functor_prod_flip C D) = functor.id :=
|
||
begin
|
||
fapply functor_eq, {intro p, apply prod.eta},
|
||
intro p p' h, cases p with c d, cases p' with c' d',
|
||
apply id_leftright,
|
||
end
|
||
end functor
|
||
open functor
|
||
|
||
namespace yoneda
|
||
--should this be defined as "yoneda_embedding Cᵒᵖ"?
|
||
definition contravariant_yoneda_embedding (C : Precategory) : Cᵒᵖ ⇒ set ^c C :=
|
||
functor_curry !hom_functor
|
||
|
||
definition yoneda_embedding (C : Precategory) : C ⇒ set ^c Cᵒᵖ :=
|
||
functor_curry (!hom_functor ∘f !functor_prod_flip)
|
||
|
||
end yoneda
|