93 lines
3.6 KiB
Text
93 lines
3.6 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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Author: Floris van Doorn
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-/
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--note: modify definition in category.set
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import .constructions .morphism
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open eq precategory equiv is_equiv is_trunc
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open is_trunc.trunctype funext precategory.ops prod.ops
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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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definition representable_functor_assoc [C : Precategory] {a1 a2 a3 a4 a5 a6 : C} (f1 : a5 ⟶ a6) (f2 : a4 ⟶ a5) (f3 : a3 ⟶ a4) (f4 : a2 ⟶ a3) (f5 : a1 ⟶ a2) : (f1 ∘ f2) ∘ f3 ∘ (f4 ∘ f5) = f1 ∘ (f2 ∘ f3 ∘ f4) ∘ f5 :=
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calc
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(f1 ∘ f2) ∘ f3 ∘ f4 ∘ f5 = 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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... = f1 ∘ (f2 ∘ f3 ∘ f4) ∘ f5 : assoc
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--disturbing behaviour: giving the type of f "(x ⟶ y)" explicitly makes the unifier loop
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definition representable_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 (λ(x : Cᵒᵖ ×c C), eq_of_homotopy (λ(h : homset x.1 x.2), !id_left ⬝ !id_right)) qed
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-- (λ(x y z : Cᵒᵖ ×c C) (g : y ⟶ z) (f : x ⟶ y), eq_of_homotopy (λ(h : hom x.1 x.2), representable_functor_assoc g.2 f.2 h f.1 g.1))
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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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attribute precategory_functor [instance] [reducible]
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namespace nat_trans
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open morphism functor
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variables {C D : Precategory} {F G : C ⇒ D} (η : F ⟹ G) (H : Π(a : C), is_iso (η a))
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include H
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definition nat_trans_inverse : G ⟹ F :=
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nat_trans.mk
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(λc, (η c)⁻¹)
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(λc d f,
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begin
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apply iso.con_inv_eq_of_eq_con,
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apply concat, rotate_left 1, apply assoc,
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apply iso.eq_inv_con_of_con_eq,
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apply inverse,
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apply naturality,
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end)
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definition nat_trans_left_inverse : nat_trans_inverse η H ∘ η = nat_trans.id :=
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begin
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fapply (apD011 nat_trans.mk),
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apply eq_of_homotopy, intro c, apply inverse_compose,
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apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, fapply is_hset.elim
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end
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definition nat_trans_right_inverse : η ∘ nat_trans_inverse η H = nat_trans.id :=
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begin
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fapply (apD011 nat_trans.mk),
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apply eq_of_homotopy, intro c, apply compose_inverse,
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apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, fapply is_hset.elim
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end
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definition nat_trans_is_iso.mk : is_iso η :=
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is_iso.mk (nat_trans_left_inverse η H) (nat_trans_right_inverse η H)
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end nat_trans
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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 : (representable_functor D) ∘f (prod_functor (opposite_functor F) (functor.ID D)) ⟹
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-- (representable_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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