40 lines
1.3 KiB
Text
40 lines
1.3 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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import ..precategory.basic ..precategory.morphism ..precategory.iso
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import hott.equiv hott.trunc
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open precategory morphism is_equiv path truncation nat sigma sigma.ops
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-- A category is a precategory extended by a witness,
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-- that the function assigning to each isomorphism a path,
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-- is an equivalecnce.
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structure category [class] (ob : Type) extends (precategory ob) :=
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(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))
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namespace category
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variables {ob : Type} {C : category ob} {a b : 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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instance [persistent] iso_of_path_equiv
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definition path_of_iso {a b : ob} : a ≅ b → a ≈ b :=
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iso_of_path⁻¹
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definition ob_1_type : is_trunc nat.zero .+1 ob :=
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begin
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apply is_trunc_succ, intros (a, b),
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fapply trunc_equiv,
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exact (@path_of_iso _ _ a b),
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apply inv_closed,
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apply is_hset_iso,
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end
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end category
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-- Bundled version of categories
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inductive Category : Type := mk : Π (ob : Type), category ob → Category
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