lean2/hott/algebra/category/constructions/hset.hlean

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/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jakob von Raumer
Category of hsets
-/
import ..category types.equiv
--open eq is_trunc sigma equiv iso is_equiv
open eq category equiv iso is_equiv is_trunc function sigma
namespace category
definition precategory_hset [reducible] : precategory hset :=
precategory.mk (λx y : hset, x → y)
(λx y z g f a, g (f a))
(λx a, a)
(λx y z w h g f, eq_of_homotopy (λa, idp))
(λx y f, eq_of_homotopy (λa, idp))
(λx y f, eq_of_homotopy (λa, idp))
definition Precategory_hset [reducible] : Precategory :=
Precategory.mk hset precategory_hset
namespace set
local attribute is_equiv_subtype_eq [instance]
definition iso_of_equiv {A B : Precategory_hset} (f : A ≃ B) : A ≅ B :=
iso.MK (to_fun f)
(equiv.to_inv f)
(eq_of_homotopy (left_inv (to_fun f)))
(eq_of_homotopy (right_inv (to_fun f)))
definition equiv_of_iso {A B : Precategory_hset} (f : A ≅ B) : A ≃ B :=
equiv.MK (to_hom f)
(iso.to_inv f)
(ap10 (right_inverse (to_hom f)))
(ap10 (left_inverse (to_hom f)))
definition is_equiv_iso_of_equiv (A B : Precategory_hset) : is_equiv (@iso_of_equiv A B) :=
adjointify _ (λf, equiv_of_iso f)
(λf, proof iso_eq idp qed)
(λf, equiv_eq idp)
local attribute is_equiv_iso_of_equiv [instance]
open sigma.ops
definition subtype_eq_inv {A : Type} {B : A → Type} [H : Πa, is_hprop (B a)] (u v : Σa, B a)
: u = v → u.1 = v.1 :=
(subtype_eq u v)⁻¹ᶠ
local attribute subtype_eq_inv [reducible]
definition is_equiv_subtype_eq_inv {A : Type} {B : A → Type} [H : Πa, is_hprop (B a)] (u v : Σa, B a)
: is_equiv (subtype_eq_inv u v) :=
_
definition iso_of_eq_eq_compose (A B : hset) : @iso_of_eq _ _ A B =
@iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
@ap _ _ (to_fun (trunctype.sigma_char 0)) A B :=
eq_of_homotopy (λp, eq.rec_on p idp)
definition equiv_equiv_iso (A B : Precategory_hset) : (A ≃ B) ≃ (A ≅ B) :=
equiv.MK (λf, iso_of_equiv f)
(λf, equiv.MK (to_hom f)
(iso.to_inv f)
(ap10 (right_inverse (to_hom f)))
(ap10 (left_inverse (to_hom f))))
(λf, iso_eq idp)
(λf, equiv_eq idp)
definition equiv_eq_iso (A B : Precategory_hset) : (A ≃ B) = (A ≅ B) :=
ua !equiv_equiv_iso
definition is_univalent_hset (A B : Precategory_hset) : is_equiv (iso_of_eq : A = B → A ≅ B) :=
have H : is_equiv (@iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
@ap _ _ (to_fun (trunctype.sigma_char 0)) A B), from
@is_equiv_compose _ _ _ _ _
(@is_equiv_compose _ _ _ _ _
(@is_equiv_compose _ _ _ _ _
_
(@is_equiv_subtype_eq_inv _ _ _ _ _))
!univalence)
!is_equiv_iso_of_equiv,
(iso_of_eq_eq_compose A B)⁻¹ ▸ H
end set
definition category_hset [instance] : category hset :=
category.mk precategory_hset set.is_univalent_hset
definition Category_hset [reducible] : Category :=
Category.mk hset category_hset
abbreviation set := Category_hset
end category