lean2/hott/algebra/category/basic.hlean

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/-
Copyright (c) 2014 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
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Module: algebra.category.basic
Author: Jakob von Raumer
-/
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import algebra.precategory.iso
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open iso is_equiv eq is_trunc
-- A category is a precategory extended by a witness
-- that the function from paths to isomorphisms,
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-- is an equivalecnce.
namespace category
definition is_univalent [reducible] {ob : Type} (C : precategory ob) :=
Π(a b : ob), is_equiv (@iso_of_eq ob C a b)
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structure category [class] (ob : Type) extends parent : precategory ob :=
(iso_of_path_equiv : is_univalent parent)
attribute category [multiple-instances]
abbreviation iso_of_path_equiv := @category.iso_of_path_equiv
definition category.mk' [reducible] (ob : Type) (C : precategory ob)
(H : Π (a b : ob), is_equiv (@iso_of_eq ob C a b)) : category ob :=
precategory.rec_on C category.mk H
section basic
variables {ob : Type} [C : category ob]
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include C
-- Make iso_of_path_equiv a class instance
-- TODO: Unsafe class instance?
attribute iso_of_path_equiv [instance]
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definition eq_of_iso [reducible] {a b : ob} : a ≅ b → a = b :=
iso_of_eq⁻¹ᵉ
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set_option apply.class_instance false -- disable class instance resolution in the apply tactic
definition is_trunc_1_ob : is_trunc 1 ob :=
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begin
apply is_trunc_succ_intro, intros (a, b),
fapply is_trunc_is_equiv_closed,
exact (@eq_of_iso _ _ a b),
apply is_equiv_inv,
apply is_hset_iso,
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end
end basic
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-- Bundled version of categories
-- we don't use Category.carrier explicitly, but rather use Precategory.carrier (to_Precategory C)
structure Category : Type :=
(carrier : Type)
(struct : category carrier)
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attribute Category.struct [instance] [coercion]
-- definition objects [reducible] := Category.objects
-- definition category_instance [instance] [coercion] [reducible] := Category.category_instance
definition Category.to_Precategory [coercion] [reducible] (C : Category) : Precategory :=
Precategory.mk (Category.carrier C) C
definition category.Mk [reducible] := Category.mk
definition category.MK [reducible] (C : Precategory)
(H : is_univalent C) : Category := Category.mk C (category.mk' C C H)
definition Category.eta (C : Category) : Category.mk C C = C :=
Category.rec (λob c, idp) C
end category