225 lines
9.7 KiB
Text
225 lines
9.7 KiB
Text
/-
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Copyright (c) 2014 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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Module: algebra.precategory.yoneda
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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 algebra.category.constructions .iso
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open category eq category.ops functor prod.ops is_trunc
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set_option pp.beta true
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namespace yoneda
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set_option class.conservative false
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--TODO: why does this take so much steps? (giving more information than "assoc" hardly helps)
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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 : assoc
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... = f1 ∘ (f2 ∘ f3) ∘ f4 ∘ f5 : assoc
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... = f1 ∘ ((f2 ∘ f3) ∘ f4) ∘ f5 : assoc
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... = _ : assoc
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--disturbing behaviour: giving the type of f "(x ⟶ y)" explicitly makes the unifier loop
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definition hom_functor (C : Precategory) : Cᵒᵖ ×c C ⇒ set :=
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functor.mk (λ(x : Cᵒᵖ ×c C), homset x.1 x.2)
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(λ(x y : Cᵒᵖ ×c C) (f : _) (h : homset x.1 x.2), f.2 ∘⁅ C ⁆ (h ∘⁅ C ⁆ f.1))
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proof
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(λ(x : Cᵒᵖ ×c C), eq_of_homotopy (λ(h : homset x.1 x.2), !id_left ⬝ !id_right))
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qed
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begin
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intros (x, y, z, g, f), apply eq_of_homotopy, intro h,
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exact (representable_functor_assoc g.2 f.2 h f.1 g.1),
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end
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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] (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, proof calc
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F (id, g' ∘ g) = F (id ∘ id, g' ∘ g) : {(id_comp c)⁻¹}
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... = F ((id,g') ∘ (id, g)) : idp
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... = F (id,g') ∘ F (id, g) : respect_comp F qed)
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local abbreviation Fob := @functor_curry_ob
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definition functor_curry_hom ⦃c c' : C⦄ (f : c ⟶ c') : Fob F c ⟹ Fob F c' :=
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nat_trans.mk (λd, F (f, id))
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(λd d' g, proof 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) : {id_left f}
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... = F (f, g) : {id_right g}
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... = F (f ∘ id, g) : {(id_right f)⁻¹}
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... = F (f ∘ id, id ∘ g) : {(id_left g)⁻¹}
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... = F (f, id) ∘ F (id, g) : (respect_comp F (f, id) (id, g))⁻¹ᵖ
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qed)
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local abbreviation Fhom := @functor_curry_hom
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definition 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_mk (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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nat_trans_eq_mk (λd, calc
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natural_map (Fhom F (f' ∘ f)) d = F (f' ∘ f, id) : functor_curry_hom_def
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... = F (f' ∘ f, id ∘ id) : {(id_comp d)⁻¹}
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... = F ((f',id) ∘ (f, id)) : idp
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... = F (f',id) ∘ F (f, id) : respect_comp F
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... = natural_map ((Fhom F f') ∘ (Fhom F f)) d : idp)
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definition functor_curry [reducible] : 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 : idp
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... = id ∘ natural_map (to_fun_hom G id) p.2 : ap (λx, x ∘ _) (respect_id (to_fun_ob G p.1) p.2)
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... = id ∘ natural_map nat_trans.id p.2 : {respect_id G p.1}
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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 : idp
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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 : {respect_comp (to_fun_ob G p''.1) f'.2 f.2}
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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 : {respect_comp G f'.1 f.1}
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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) : idp
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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) : idp
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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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square_prepostcompose (!naturality⁻¹ᵖ) _ _
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... = Ghom G f' ∘ Ghom G f : idp
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definition functor_uncurry [reducible] : 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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-- open pi
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-- definition functor_eq_mk'1 {F₁ F₂ : C → D} {H₁ : Π(a b : C), hom a b → hom (F₁ a) (F₁ b)}
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-- {H₂ : Π(a b : C), hom a b → hom (F₂ a) (F₂ b)} (id₁ id₂ comp₁ comp₂)
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-- (pF : F₁ = F₂) (pH : Π(a b : C) (f : hom a b), pF ▹ (H₁ a b f) = H₂ a b f)
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-- : functor.mk F₁ H₁ id₁ comp₁ = functor.mk F₂ H₂ id₂ comp₂ :=
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-- functor_eq_mk'' id₁ id₂ comp₁ comp₂ pF
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-- (eq_of_homotopy (λc, eq_of_homotopy (λc', eq_of_homotopy (λf,
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-- begin
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-- apply concat, rotate_left 1, exact (pH c c' f),
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-- apply concat, rotate_left 1,
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-- exact (pi_transport_constant pF (H₁ c c') f),
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-- apply (apD10' f),
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-- apply concat, rotate_left 1,
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-- exact (pi_transport_constant pF (H₁ c) c'),
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-- apply (apD10' c'),
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-- apply concat, rotate_left 1,
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-- exact (pi_transport_constant pF H₁ c),
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-- apply idp
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-- end))))
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-- definition functor_eq_mk1 {F₁ F₂ : C ⇒ D} : Π(p : to_fun_ob F₁ = to_fun_ob F₂),
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-- (Π(a b : C) (f : hom a b), transport (λF, hom (F a) (F b)) p (F₁ f) = F₂ f)
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-- → F₁ = F₂ :=
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-- functor.rec_on F₁ (λO₁ H₁ id₁ comp₁, functor.rec_on F₂ (λO₂ H₂ id₂ comp₂ p, !functor_eq_mk'1))
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--set_option pp.notation false
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definition functor_uncurry_functor_curry : functor_uncurry (functor_curry F) = F :=
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functor_eq_mk (λp, ap (to_fun_ob F) !prod.eta)
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begin
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intros (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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have H : (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) : idp
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... = F (id ∘ f, g ∘ id) : respect_comp F (id,g) (f,id)
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... = F (f, g ∘ id) : {id_left f}
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... = F (f,g) : {id_right g},
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rewrite H,
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apply sorry
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end
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--set_option pp.implicit true
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definition functor_curry_functor_uncurry : functor_curry (functor_uncurry G) = G :=
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begin
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fapply functor_eq_mk,
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{intro c,
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fapply functor_eq_mk,
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{intro d, apply idp},
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{intros (d, d', g),
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have H : 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 : idp
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... = to_fun_hom (G c) g ∘ natural_map (ID (G c)) d
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: ap (λx, to_fun_hom (G c) g ∘ natural_map x d) (respect_id G c)
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... = to_fun_hom (G c) g : id_right,
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rewrite H,
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-- esimp {idp},
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apply sorry
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}
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},
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apply sorry
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end
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definition equiv_functor_curry : (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_ob : C ×c D ⇒ D ×c C :=
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functor.mk sorry sorry sorry sorry
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definition contravariant_yoneda_embedding : Cᵒᵖ ⇒ set ^c C :=
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functor_curry !yoneda.hom_functor
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end functor
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-- Coq uses unit/counit definitions as basic
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-- open yoneda precategory.product precategory.opposite functor morphism
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-- --universe levels are given explicitly because Lean uses 6 variables otherwise
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-- structure adjoint.{u v} [C D : Precategory.{u v}] (F : C ⇒ D) (G : D ⇒ C) : Type.{max u v} :=
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-- (nat_iso : (hom_functor D) ∘f (prod_functor (opposite_functor F) (functor.ID D)) ⟹
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-- (hom_functor C) ∘f (prod_functor (functor.ID (Cᵒᵖ)) G))
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-- (is_iso_nat_iso : is_iso nat_iso)
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-- infix `⊣`:55 := adjoint
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-- namespace adjoint
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-- universe variables l1 l2
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-- variables [C D : Precategory.{l1 l2}] (F : C ⇒ D) (G : D ⇒ C)
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-- end adjoint
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