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