lean2/library/hott/algebra/category/basic.lean

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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
import hott.equiv hott.trunc
open precategory morphism is_equiv path 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))
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?
instance [persistent] iso_of_path_equiv
definition path_of_iso {a b : ob} : a ≅ b → a ≈ b :=
iso_of_path⁻¹
definition foo {a b : ob} :
definition ob_1_type : is_trunc -2 .+1 .+1 .+1 ob :=
begin
apply is_trunc_succ, intros (a, b),
/-fapply trunc_equiv,
exact (@path_of_iso ob C a b),
apply inv_closed,
exact sorry,-/
end
end category
-- Bundled version of categories
inductive Category : Type := mk : Π (ob : Type), category ob → Category