83 lines
2.7 KiB
Text
83 lines
2.7 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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Author: Jakob von Raumer
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-/
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import .iso
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open iso is_equiv 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 : a = b → a ≅ b)
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structure category [class] (ob : Type) extends parent : precategory ob :=
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mk' :: (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] [unfold 2] {ob : Type} (C : precategory ob)
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(H : Π (a b : ob), is_equiv (iso_of_eq : a = b → 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_equiv_iso [constructor] (a b : ob) : (a = b) ≃ (a ≅ b) :=
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equiv.mk iso_of_eq _
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definition eq_of_iso [reducible] {a b : ob} : a ≅ b → a = b :=
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iso_of_eq⁻¹ᶠ
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definition iso_of_eq_eq_of_iso {a b : ob} (p : a ≅ b) : iso_of_eq (eq_of_iso p) = p :=
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right_inv iso_of_eq p
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definition hom_of_eq_eq_of_iso {a b : ob} (p : a ≅ b) : hom_of_eq (eq_of_iso p) = to_hom p :=
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ap to_hom !iso_of_eq_eq_of_iso
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definition inv_of_eq_eq_of_iso {a b : ob} (p : a ≅ b) : inv_of_eq (eq_of_iso p) = to_inv p :=
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ap to_inv !iso_of_eq_eq_of_iso
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theorem eq_of_iso_refl {a : ob} : eq_of_iso (iso.refl a) = idp :=
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inv_eq_of_eq idp
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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, intro 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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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 Category.to_Precategory [constructor] [coercion] [reducible] (C : Category)
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: Precategory :=
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Precategory.mk (Category.carrier C) C
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definition category.Mk [constructor] [reducible] := Category.mk
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definition category.MK [constructor] [reducible] (C : Precategory)
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(H : is_univalent C) : Category := Category.mk C (category.mk 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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