206 lines
9 KiB
Text
206 lines
9 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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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 (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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begin
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intro x, apply eq_of_homotopy, intro h, exact (!id_left ⬝ !id_right)
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end
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begin
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intro 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, 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 ⦃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, 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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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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nat_trans_eq (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) : 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) : respect_comp F
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... = natural_map ((Fhom F f') ∘ (Fhom F f)) d : by esimp)
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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 : 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 ∘ 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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square_prepostcompose (!naturality⁻¹ᵖ) _ _
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... = Ghom G f' ∘ Ghom G f : by esimp
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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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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) : 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 : 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 (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 (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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--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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end yoneda
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