159 lines
6.7 KiB
Text
159 lines
6.7 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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Yoneda embedding and Yoneda lemma
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-/
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import .functor.curry .constructions.hset .constructions.opposite .functor.attributes
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open category eq functor prod.ops is_trunc iso is_equiv equiv category.set nat_trans lift
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namespace yoneda
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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)
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(h : @homset (Cᵒᵖ) C x.1 x.2), f.2 ∘[C] (h ∘[C] f.1))
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(λ x, abstract @eq_of_homotopy _ _ _ (ID (@homset Cᵒᵖ C x.1 x.2))
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(λ h, concat (by apply @id_left) (by apply @id_right)) end)
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(λ x y z g f, abstract eq_of_homotopy (by intros; apply @representable_functor_assoc) end)
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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 [reducible] (C : Precategory) : Cᵒᵖ ⇒ cset ^c C :=
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functor_curry !hom_functor
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definition yoneda_embedding (C : Precategory) : C ⇒ cset ^c Cᵒᵖ :=
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functor_curry (!hom_functor ∘f !prod_flip_functor)
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notation `ɏ` := yoneda_embedding _
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definition yoneda_lemma_hom [constructor] {C : Precategory} (c : C) (F : Cᵒᵖ ⇒ cset)
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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 cset)⁻¹ x}
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end
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definition yoneda_lemma_equiv [constructor] {C : Precategory} (c : C)
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(F : Cᵒᵖ ⇒ cset) : hom (ɏ c) F ≃ lift (F c) :=
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begin
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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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definition yoneda_lemma {C : Precategory} (c : C) (F : Cᵒᵖ ⇒ cset) :
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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, apply yoneda_lemma_equiv,
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end
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theorem yoneda_lemma_natural_ob {C : Precategory} (F : Cᵒᵖ ⇒ cset) {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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to_hom (yoneda_lemma c' F) (η ∘n to_fun_hom ɏ f) :=
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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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-- TODO: Investigate what is the bottleneck to type check the next theorem
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-- attribute yoneda_lemma lift_functor Precategory_hset precategory_hset homset
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-- yoneda_embedding nat_trans.compose functor_nat_trans_compose [reducible]
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-- attribute tlift functor.compose [reducible]
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theorem yoneda_lemma_natural_functor.{u v} {C : Precategory.{u v}} (c : C) (F F' : Cᵒᵖ ⇒ cset)
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(θ : F ⟹ F') (η : to_fun_ob ɏ c ⟹ F) :
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(lift_functor.{v u} ∘fn θ) c (to_hom (yoneda_lemma c F) η) =
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proof to_hom (yoneda_lemma c F') (θ ∘n η) qed :=
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by reflexivity
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-- theorem xx.{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 _ qed =
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-- to_hom (yoneda_lemma c F') (θ ∘n η) :=
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-- by reflexivity
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-- theorem yy.{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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-- (lift_functor.{v u} ∘fn θ) c (to_hom (yoneda_lemma c F) η) =
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-- proof _ 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 ⇒ cset ^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 is_embedding_yoneda_embedding (C : Category) :
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is_embedding (ɏ : C → Cᵒᵖ ⇒ cset) :=
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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, to_fun_iso], -- to_fun_iso not unfolded
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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 ⇒ cset) := Σ(c : C), ɏ c ≅ F
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section
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set_option apply.class_instance false
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definition is_hprop_representable {C : Category} (F : Cᵒᵖ ⇒ cset)
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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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{ exact proof fiber.sigma_char ɏ F qed ⬝e sigma.sigma_equiv_sigma_id (λc, !eq_equiv_iso)},
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{ apply function.is_hprop_fiber_of_is_embedding, apply is_embedding_yoneda_embedding}
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end
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end
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end yoneda
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