141 lines
5.2 KiB
Text
141 lines
5.2 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, Jakob von Raumer
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Category of hsets
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-/
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import ..category types.equiv ..functor types.lift ..limits
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open eq category equiv iso is_equiv is_trunc function sigma
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namespace category
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definition precategory_hset.{u} [reducible] [constructor] : precategory hset.{u} :=
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precategory.mk (λx y : hset, x → y)
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(λx y z g f a, g (f a))
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(λx a, a)
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(λx y z w h g f, eq_of_homotopy (λa, idp))
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(λx y f, eq_of_homotopy (λa, idp))
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(λx y f, eq_of_homotopy (λa, idp))
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definition Precategory_hset [reducible] [constructor] : Precategory :=
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Precategory.mk hset precategory_hset
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namespace set
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local attribute is_equiv_subtype_eq [instance]
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definition iso_of_equiv [constructor] {A B : Precategory_hset} (f : A ≃ B) : A ≅ B :=
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iso.MK (to_fun f)
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(to_inv f)
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(eq_of_homotopy (left_inv (to_fun f)))
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(eq_of_homotopy (right_inv (to_fun f)))
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definition equiv_of_iso [constructor] {A B : Precategory_hset} (f : A ≅ B) : A ≃ B :=
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begin
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apply equiv.MK (to_hom f) (iso.to_inv f),
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exact ap10 (to_right_inverse f),
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exact ap10 (to_left_inverse f)
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end
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definition is_equiv_iso_of_equiv [constructor] (A B : Precategory_hset)
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: is_equiv (@iso_of_equiv A B) :=
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adjointify _ (λf, equiv_of_iso f)
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(λf, proof iso_eq idp qed)
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(λf, equiv_eq idp)
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local attribute is_equiv_iso_of_equiv [instance]
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definition iso_of_eq_eq_compose (A B : hset) : @iso_of_eq _ _ A B =
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@iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
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@ap _ _ (to_fun (trunctype.sigma_char 0)) A B :=
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eq_of_homotopy (λp, eq.rec_on p idp)
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definition equiv_equiv_iso (A B : Precategory_hset) : (A ≃ B) ≃ (A ≅ B) :=
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equiv.MK (λf, iso_of_equiv f)
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(λf, proof equiv.MK (to_hom f)
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(iso.to_inv f)
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(ap10 (to_right_inverse f))
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(ap10 (to_left_inverse f)) qed)
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(λf, proof iso_eq idp qed)
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(λf, proof equiv_eq idp qed)
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definition equiv_eq_iso (A B : Precategory_hset) : (A ≃ B) = (A ≅ B) :=
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ua !equiv_equiv_iso
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definition is_univalent_hset (A B : Precategory_hset) : is_equiv (iso_of_eq : A = B → A ≅ B) :=
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assert H₁ : is_equiv (@iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
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@ap _ _ (to_fun (trunctype.sigma_char 0)) A B), from
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@is_equiv_compose _ _ _ _ _
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(@is_equiv_compose _ _ _ _ _
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(@is_equiv_compose _ _ _ _ _
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_
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(@is_equiv_subtype_eq_inv _ _ _ _ _))
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!univalence)
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!is_equiv_iso_of_equiv,
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let H₂ := (iso_of_eq_eq_compose A B)⁻¹ in
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begin
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rewrite H₂ at H₁,
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assumption
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end
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end set
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definition category_hset [instance] [constructor] : category hset :=
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category.mk precategory_hset set.is_univalent_hset
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definition Category_hset [reducible] [constructor] : Category :=
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Category.mk hset category_hset
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abbreviation set [constructor] := Category_hset
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open functor lift
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definition lift_functor.{u v} [constructor] : set.{u} ⇒ set.{max u v} :=
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functor.mk tlift
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(λa b, lift_functor)
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(λa, eq_of_homotopy (λx, by induction x; reflexivity))
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(λa b c g f, eq_of_homotopy (λx, by induction x; reflexivity))
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open pi sigma.ops
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local attribute Category.to.precategory [unfold 1]
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local attribute category.to_precategory [unfold 2]
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definition is_complete_set_cone.{u v w} [constructor]
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(I : Precategory.{v w}) (F : I ⇒ set.{max u v w}) : cone_obj F :=
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begin
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fapply cone_obj.mk,
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{ fapply trunctype.mk,
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{ exact Σ(s : Π(i : I), trunctype.carrier (F i)),
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Π{i j : I} (f : i ⟶ j), F f (s i) = (s j)},
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{ exact abstract begin apply is_trunc_sigma, intro s,
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apply is_trunc_pi, intro i,
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apply is_trunc_pi, intro j,
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apply is_trunc_pi, intro f,
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apply is_trunc_eq end end}},
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{ fapply nat_trans.mk,
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{ intro i x, esimp at x, exact x.1 i},
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{ intro i j f, esimp, apply eq_of_homotopy, intro x, esimp at x, induction x with s p,
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esimp, apply p}}
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end
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definition is_complete_set.{u v w} [instance] : is_complete.{(max u v w)+1 (max u v w) v w} set :=
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begin
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intro I F, fapply has_terminal_object.mk,
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{ exact is_complete_set_cone.{u v w} I F},
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{ intro c, esimp at *, induction c with X η, induction η with η p, esimp at *,
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fapply is_contr.mk,
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{ fapply cone_hom.mk,
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{ intro x, esimp at *, fapply sigma.mk,
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{ intro i, exact η i x},
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{ intro i j f, exact ap10 (p f) x}},
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{ intro i, reflexivity}},
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{ esimp, intro h, induction h with f q, apply cone_hom_eq, esimp at *,
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apply eq_of_homotopy, intro x, fapply sigma_eq: esimp,
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{ apply eq_of_homotopy, intro i, exact (ap10 (q i) x)⁻¹},
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{ apply is_hprop.elimo,
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apply is_trunc_pi, intro i,
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apply is_trunc_pi, intro j,
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apply is_trunc_pi, intro f,
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apply is_trunc_eq}}}
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end
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end category
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