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.
-- Author: Jakob von Raumer
import ..precategory.basic ..precategory.morphism ..precategory.iso
open precategory morphism is_equiv eq truncation 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
definition ob_1_type : is_trunc nat.zero .+1 ob :=
begin
apply is_trunc_succ, intros (a, b),
fapply trunc_equiv,
exact (@path_of_iso _ _ a b),
apply inv_closed,
apply is_hset_iso,
end
end category
-- Bundled version of categories
inductive Category : Type := mk : Π (ob : Type), category ob → Category