305 lines
13 KiB
Text
305 lines
13 KiB
Text
/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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-/
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--note: modify definition in category.set
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import .constructions.functor .constructions.hset .constructions.product .constructions.opposite
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.adjoint
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open category eq category.ops functor prod.ops is_trunc iso
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namespace yoneda
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-- set_option class.conservative false
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definition representable_functor_assoc [C : Precategory] {a1 a2 a3 a4 a5 a6 : C}
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(f1 : hom a5 a6) (f2 : hom a4 a5) (f3 : hom a3 a4) (f4 : hom a2 a3) (f5 : hom a1 a2)
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: (f1 ∘ f2) ∘ f3 ∘ (f4 ∘ f5) = f1 ∘ (f2 ∘ f3 ∘ f4) ∘ f5 :=
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calc
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_ = f1 ∘ f2 ∘ f3 ∘ f4 ∘ f5 : by rewrite -assoc
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... = f1 ∘ (f2 ∘ f3) ∘ f4 ∘ f5 : by rewrite -assoc
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... = f1 ∘ ((f2 ∘ f3) ∘ f4) ∘ f5 : by rewrite -(assoc (f2 ∘ f3) _ _)
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... = _ : by rewrite (assoc f2 f3 f4)
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definition hom_functor.{u v} [constructor] (C : Precategory.{u v}) : Cᵒᵖ ×c C ⇒ set.{v} :=
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functor.mk
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(λ (x : Cᵒᵖ ×c C), @homset (Cᵒᵖ) C x.1 x.2)
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(λ (x y : Cᵒᵖ ×c C) (f : @category.precategory.hom (Cᵒᵖ ×c C) (Cᵒᵖ ×c C) x y) (h : @homset (Cᵒᵖ) C x.1 x.2),
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f.2 ∘[C] (h ∘[C] f.1))
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(λ x, @eq_of_homotopy _ _ _ (ID (@homset Cᵒᵖ C x.1 x.2)) (λ h, concat (by apply @id_left) (by apply @id_right)))
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(λ x y z g f,
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eq_of_homotopy (by intros; apply @representable_functor_assoc))
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end yoneda
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open is_equiv equiv
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namespace functor
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open prod nat_trans
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variables {C D E : Precategory} (F : C ×c D ⇒ E) (G : C ⇒ E ^c D)
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definition functor_curry_ob [reducible] [constructor] (c : C) : E ^c D :=
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functor.mk (λd, F (c,d))
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(λd d' g, F (id, g))
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(λd, !respect_id)
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(λd₁ d₂ d₃ g' g, calc
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F (id, g' ∘ g) = F (id ∘ id, g' ∘ g) : by rewrite id_comp
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... = F ((id,g') ∘ (id, g)) : by esimp
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... = F (id,g') ∘ F (id, g) : by rewrite respect_comp)
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local abbreviation Fob := @functor_curry_ob
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definition functor_curry_hom [constructor] ⦃c c' : C⦄ (f : c ⟶ c') : Fob F c ⟹ Fob F c' :=
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begin
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fapply @nat_trans.mk,
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{intro d, exact F (f, id)},
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{intro d d' g, calc
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F (id, g) ∘ F (f, id) = F (id ∘ f, g ∘ id) : respect_comp F
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... = F (f, g ∘ id) : by rewrite id_left
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... = F (f, g) : by rewrite id_right
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... = F (f ∘ id, g) : by rewrite id_right
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... = F (f ∘ id, id ∘ g) : by rewrite id_left
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... = F (f, id) ∘ F (id, g) : (respect_comp F (f, id) (id, g))⁻¹ᵖ
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}
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end
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local abbreviation Fhom := @functor_curry_hom
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theorem functor_curry_hom_def ⦃c c' : C⦄ (f : c ⟶ c') (d : D) :
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(Fhom F f) d = to_fun_hom F (f, id) := idp
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theorem functor_curry_id (c : C) : Fhom F (ID c) = nat_trans.id :=
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nat_trans_eq (λd, respect_id F _)
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theorem functor_curry_comp ⦃c c' c'' : C⦄ (f' : c' ⟶ c'') (f : c ⟶ c')
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: Fhom F (f' ∘ f) = Fhom F f' ∘n Fhom F f :=
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begin
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apply @nat_trans_eq,
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intro d, calc
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natural_map (Fhom F (f' ∘ f)) d = F (f' ∘ f, id) : by rewrite functor_curry_hom_def
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... = F (f' ∘ f, id ∘ id) : by rewrite id_comp
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... = F ((f',id) ∘ (f, id)) : by esimp
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... = F (f',id) ∘ F (f, id) : by rewrite [respect_comp F]
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... = natural_map ((Fhom F f') ∘ (Fhom F f)) d : by esimp
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end
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definition functor_curry [reducible] [constructor] : C ⇒ E ^c D :=
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functor.mk (functor_curry_ob F)
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(functor_curry_hom F)
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(functor_curry_id F)
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(functor_curry_comp F)
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definition functor_uncurry_ob [reducible] (p : C ×c D) : E :=
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to_fun_ob (G p.1) p.2
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local abbreviation Gob := @functor_uncurry_ob
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definition functor_uncurry_hom ⦃p p' : C ×c D⦄ (f : hom p p') : Gob G p ⟶ Gob G p' :=
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to_fun_hom (to_fun_ob G p'.1) f.2 ∘ natural_map (to_fun_hom G f.1) p.2
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local abbreviation Ghom := @functor_uncurry_hom
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theorem functor_uncurry_id (p : C ×c D) : Ghom G (ID p) = id :=
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calc
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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
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... = id ∘ natural_map (to_fun_hom G id) p.2 : by rewrite respect_id
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... = id ∘ natural_map nat_trans.id p.2 : by rewrite respect_id
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... = id : id_comp
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theorem functor_uncurry_comp ⦃p p' p'' : C ×c D⦄ (f' : p' ⟶ p'') (f : p ⟶ p')
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: Ghom G (f' ∘ f) = Ghom G f' ∘ Ghom G f :=
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calc
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Ghom G (f' ∘ f)
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= 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
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
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∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : by rewrite respect_comp
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
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∘ natural_map (to_fun_hom G f'.1 ∘ to_fun_hom G f.1) p.2 : by rewrite respect_comp
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
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∘ (natural_map (to_fun_hom G f'.1) p.2 ∘ natural_map (to_fun_hom G f.1) p.2) : by esimp
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ natural_map (to_fun_hom G f'.1) p'.2)
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∘ (to_fun_hom (to_fun_ob G p'.1) f.2 ∘ natural_map (to_fun_hom G f.1) p.2) :
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by rewrite [square_prepostcompose (!naturality⁻¹ᵖ) _ _]
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... = Ghom G f' ∘ Ghom G f : by esimp
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definition functor_uncurry [reducible] [constructor] : C ×c D ⇒ E :=
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functor.mk (functor_uncurry_ob G)
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(functor_uncurry_hom G)
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(functor_uncurry_id G)
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(functor_uncurry_comp G)
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theorem functor_uncurry_functor_curry : functor_uncurry (functor_curry F) = F :=
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functor_eq (λp, ap (to_fun_ob F) !prod.eta)
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begin
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intro cd cd' fg,
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cases cd with c d, cases cd' with c' d', cases fg with f g,
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transitivity to_fun_hom (functor_uncurry (functor_curry F)) (f, g),
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apply id_leftright,
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show (functor_uncurry (functor_curry F)) (f, g) = F (f,g),
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from calc
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(functor_uncurry (functor_curry F)) (f, g) = to_fun_hom F (id, g) ∘ to_fun_hom F (f, id) : by esimp
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... = F (id ∘ f, g ∘ id) : by krewrite [-respect_comp F (id,g) (f,id)]
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... = F (f, g ∘ id) : by rewrite id_left
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... = F (f,g) : by rewrite id_right,
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end
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definition functor_curry_functor_uncurry_ob (c : C)
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: functor_curry (functor_uncurry G) c = G c :=
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begin
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fapply functor_eq,
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{intro d, reflexivity},
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{intro d d' g,
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apply concat, apply id_leftright,
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show to_fun_hom (functor_curry (functor_uncurry G) c) g = to_fun_hom (G c) g,
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from calc
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to_fun_hom (functor_curry (functor_uncurry G) c) g
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= to_fun_hom (G c) g ∘ natural_map (to_fun_hom G (ID c)) d : by esimp
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... = to_fun_hom (G c) g ∘ natural_map (ID (G c)) d : by rewrite respect_id
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... = to_fun_hom (G c) g ∘ id : by reflexivity
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... = to_fun_hom (G c) g : by rewrite id_right}
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end
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theorem functor_curry_functor_uncurry : functor_curry (functor_uncurry G) = G :=
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begin
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fapply functor_eq, exact (functor_curry_functor_uncurry_ob G),
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intro c c' f,
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fapply nat_trans_eq,
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intro d,
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apply concat,
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{apply (ap (λx, x ∘ _)),
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apply concat, apply natural_map_hom_of_eq, apply (ap hom_of_eq), apply ap010_functor_eq},
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apply concat,
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{apply (ap (λx, _ ∘ x)), apply (ap (λx, _ ∘ x)),
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apply concat, apply natural_map_inv_of_eq,
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apply (ap (λx, hom_of_eq x⁻¹)), apply ap010_functor_eq},
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apply concat, apply id_leftright,
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apply concat, apply (ap (λx, x ∘ _)), apply respect_id,
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apply id_left
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end
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definition prod_functor_equiv_functor_functor [constructor] (C D E : Precategory)
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: (C ×c D ⇒ E) ≃ (C ⇒ E ^c D) :=
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equiv.MK functor_curry
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functor_uncurry
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functor_curry_functor_uncurry
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functor_uncurry_functor_curry
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definition functor_prod_flip [constructor] (C D : Precategory) : C ×c D ⇒ D ×c C :=
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functor.mk (λp, (p.2, p.1))
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(λp p' h, (h.2, h.1))
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(λp, idp)
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(λp p' p'' h' h, idp)
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definition functor_prod_flip_functor_prod_flip (C D : Precategory)
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: functor_prod_flip D C ∘f (functor_prod_flip C D) = functor.id :=
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begin
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fapply functor_eq, {intro p, apply prod.eta},
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intro p p' h, cases p with c d, cases p' with c' d',
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apply id_leftright,
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end
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end functor
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open functor
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namespace yoneda
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open category.set nat_trans lift
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/-
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These attributes make sure that the fields of the category "set" reduce to the right things
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However, we don't want to have them globally, because that will unfold the composition g ∘ f
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in a Category to category.category.comp g f
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-/
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local attribute Category.to.precategory category.to_precategory [constructor]
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-- should this be defined as "yoneda_embedding Cᵒᵖ"?
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definition contravariant_yoneda_embedding (C : Precategory) : Cᵒᵖ ⇒ set ^c C :=
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functor_curry !hom_functor
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definition yoneda_embedding (C : Precategory) : C ⇒ set ^c Cᵒᵖ :=
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functor_curry (!hom_functor ∘f !functor_prod_flip)
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notation `ɏ` := yoneda_embedding _
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definition yoneda_lemma_hom [constructor] {C : Precategory} (c : C) (F : Cᵒᵖ ⇒ set)
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(x : trunctype.carrier (F c)) : ɏ c ⟹ F :=
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begin
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fapply nat_trans.mk,
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{ intro c', esimp [yoneda_embedding], intro f, exact F f x},
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{ intro c' c'' f, esimp [yoneda_embedding], apply eq_of_homotopy, intro f',
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refine _ ⬝ ap (λy, to_fun_hom F y x) !(@id_left _ C)⁻¹,
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exact ap10 !(@respect_comp Cᵒᵖ set)⁻¹ x}
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end
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definition yoneda_lemma {C : Precategory} (c : C) (F : Cᵒᵖ ⇒ set) :
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homset (ɏ c) F ≅ lift_functor (F c) :=
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begin
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apply iso_of_equiv, esimp,
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fapply equiv.MK,
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{ intro η, exact up (η c id)},
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{ intro x, induction x with x, exact yoneda_lemma_hom c F x},
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{ exact abstract begin intro x, induction x with x, esimp, apply ap up,
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exact ap10 !respect_id x end end},
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{ exact abstract begin intro η, esimp, apply nat_trans_eq,
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intro c', esimp, apply eq_of_homotopy,
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intro f, esimp [yoneda_embedding] at f,
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transitivity (F f ∘ η c) id, reflexivity,
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rewrite naturality, esimp [yoneda_embedding], rewrite [id_left], apply ap _ !id_left end end},
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end
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theorem yoneda_lemma_natural_ob {C : Precategory} (F : Cᵒᵖ ⇒ set) {c c' : C} (f : c' ⟶ c)
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(η : ɏ c ⟹ F) :
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to_fun_hom (lift_functor ∘f F) f (to_hom (yoneda_lemma c F) η) =
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proof to_hom (yoneda_lemma c' F) (η ∘n to_fun_hom ɏ f) qed :=
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begin
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esimp [yoneda_lemma,yoneda_embedding], apply ap up,
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transitivity (F f ∘ η c) id, reflexivity,
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rewrite naturality,
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esimp [yoneda_embedding],
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apply ap (η c'),
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esimp [yoneda_embedding, Opposite],
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rewrite [+id_left,+id_right],
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end
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theorem yoneda_lemma_natural_functor.{u v} {C : Precategory.{u v}} (c : C) (F F' : Cᵒᵖ ⇒ set)
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(θ : F ⟹ F') (η : to_fun_ob ɏ c ⟹ F) :
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proof (lift_functor.{v u} ∘fn θ) c (to_hom (yoneda_lemma c F) η) qed =
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(to_hom (yoneda_lemma c F') proof (θ ∘n η : (to_fun_ob ɏ c : Cᵒᵖ ⇒ set) ⟹ F') qed) :=
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by reflexivity
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definition fully_faithful_yoneda_embedding [instance] (C : Precategory) :
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fully_faithful (ɏ : C ⇒ set ^c Cᵒᵖ) :=
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begin
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intro c c',
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fapply is_equiv_of_equiv_of_homotopy,
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{ symmetry, transitivity _, apply @equiv_of_iso (homset _ _),
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rexact yoneda_lemma c (ɏ c'), esimp [yoneda_embedding], exact !equiv_lift⁻¹ᵉ},
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{ intro f, apply nat_trans_eq, intro c, apply eq_of_homotopy, intro f',
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esimp [equiv.symm,equiv.trans],
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esimp [yoneda_lemma,yoneda_embedding,Opposite],
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rewrite [id_left,id_right]}
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end
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definition embedding_on_objects_yoneda_embedding (C : Category) :
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is_embedding (ɏ : C → Cᵒᵖ ⇒ set) :=
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begin
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intro c c', fapply is_equiv_of_equiv_of_homotopy,
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{ exact !eq_equiv_iso ⬝e !iso_equiv_F_iso_F ⬝e !eq_equiv_iso⁻¹ᵉ},
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{ intro p, induction p, esimp [equiv.trans, equiv.symm],
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esimp [to_fun_iso],
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rewrite -eq_of_iso_refl,
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apply ap eq_of_iso, apply iso_eq, esimp,
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apply nat_trans_eq, intro c',
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apply eq_of_homotopy, esimp [yoneda_embedding], intro f,
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rewrite [category.category.id_left], apply id_right}
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end
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definition is_representable {C : Precategory} (F : Cᵒᵖ ⇒ set) := Σ(c : C), ɏ c ≅ F
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definition is_hprop_representable {C : Category} (F : Cᵒᵖ ⇒ set)
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: is_hprop (is_representable F) :=
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begin
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fapply is_trunc_equiv_closed,
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{ transitivity _, rotate 1,
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{ apply sigma.sigma_equiv_sigma_id, intro c, exact !eq_equiv_iso},
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{ apply fiber.sigma_char}},
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{ apply function.is_hprop_fiber_of_is_embedding,
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apply embedding_on_objects_yoneda_embedding}
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end
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end yoneda
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