lean2/hott/algebra/category/basic.hlean
Floris van Doorn 61901cff81 feat(hott): rename definition and cleanup in HoTT library
also add more definitions in types.pi, types.path, algebra.precategory

the (pre)category library still needs cleanup
authors of this commit: @avigad, @javra, @fpvandoorn
2015-02-20 21:40:42 -05:00

59 lines
1.9 KiB
Text

-- Copyright (c) 2014 Jakob von Raumer. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Jakob von Raumer
import ..precategory.basic ..precategory.morphism ..precategory.iso
open precategory morphism is_equiv eq is_trunc nat sigma sigma.ops
-- A category is a precategory extended by a witness,
-- that the function assigning to each isomorphism a path,
-- is an equivalecnce.
structure category [class] (ob : Type) extends (precategory ob) :=
(iso_of_path_equiv : Π {a b : ob}, is_equiv (@iso_of_path ob (precategory.mk hom _ comp ID assoc id_left id_right) a b))
attribute category [multiple-instances]
namespace category
variables {ob : Type} {C : category ob} {a b : ob}
include C
-- Make iso_of_path_equiv a class instance
-- TODO: Unsafe class instance?
attribute iso_of_path_equiv [instance]
definition path_of_iso {a b : ob} : a ≅ b → a = b :=
iso_of_path⁻¹
set_option apply.class_instance false -- disable class instance resolution in the apply tactic
definition ob_1_type : is_trunc (succ nat.zero) ob :=
begin
apply is_trunc_succ_intro, intros (a, b),
fapply is_trunc_is_equiv_closed,
exact (@path_of_iso _ _ a b),
apply is_equiv_inv,
apply is_hset_iso,
end
end category
-- Bundled version of categories
structure Category : Type :=
(objects : Type)
(category_instance : category objects)
namespace category
definition Mk {ob} (C) : Category := Category.mk ob C
--definition MK (a b c d e f g h i) : Category := Category.mk a (category.mk b c d e f g h i)
definition objects [coercion] [reducible] := Category.objects
definition category_instance [instance] [coercion] [reducible] := Category.category_instance
end category
open category
protected definition Category.eta (C : Category) : Category.mk C C = C :=
Category.rec (λob c, idp) C