85 lines
2.8 KiB
Text
85 lines
2.8 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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Authors: Jakob von Raumer, Floris van Doorn
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Ported from Coq HoTT
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-/
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import .iso algebra.bundled
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open eq is_trunc iso category algebra nat unit
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namespace category
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structure groupoid [class] (ob : Type) extends parent : precategory ob :=
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mk' :: (all_iso : Π ⦃a b : ob⦄ (f : hom a b), @is_iso ob parent a b f)
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abbreviation all_iso := @groupoid.all_iso
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attribute groupoid.all_iso [instance] [priority 3000]
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attribute groupoid.to_precategory [unfold 2]
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definition groupoid.mk [reducible] [constructor] {ob : Type} (C : precategory ob)
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(H : Π (a b : ob) (f : a ⟶ b), is_iso f) : groupoid ob :=
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precategory.rec_on C groupoid.mk' H
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definition groupoid_of_group.{l} [constructor] (A : Type.{l}) [G : group A] :
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groupoid.{0 l} unit :=
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begin
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fapply groupoid.mk; fapply precategory.mk: intros,
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{ exact A},
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{ exact _},
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{ exact a_2 * a_1},
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{ exact 1},
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{ apply mul.assoc},
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{ apply mul_one},
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{ apply one_mul},
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{ esimp [precategory.mk],
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fapply is_iso.mk,
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{ exact f⁻¹},
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{ apply mul.right_inv},
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{ apply mul.left_inv}},
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end
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definition hom_group [constructor] {A : Type} [G : groupoid A] (a : A) : group (hom a a) :=
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begin
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fapply group.mk,
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intro f g, apply (comp f g),
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apply is_set_hom,
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intros f g h, apply (assoc f g h)⁻¹,
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apply (ID a),
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intro f, apply id_left,
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intro f, apply id_right,
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intro f, exact (iso.inverse f),
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intro f, exact (iso.left_inverse f),
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end
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definition group_of_is_contr_groupoid {ob : Type} [H : is_contr ob]
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[G : groupoid ob] : group (hom (center ob) (center ob)) := !hom_group
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definition group_of_groupoid_unit [G : groupoid unit] : group (hom ⋆ ⋆) := !hom_group
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-- Bundled version of categories
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-- we don't use Groupoid.carrier explicitly, but rather use Groupoid.carrier (to_Precategory C)
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structure Groupoid : Type :=
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(carrier : Type)
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(struct : groupoid carrier)
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attribute Groupoid.struct [instance] [coercion]
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definition Groupoid.to_Precategory [coercion] [reducible] [unfold 1] (C : Groupoid)
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: Precategory :=
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Precategory.mk (Groupoid.carrier C) _
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attribute Groupoid._trans_of_to_Precategory_1 [unfold 1]
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definition groupoid.Mk [reducible] [constructor] := Groupoid.mk
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definition groupoid.MK [reducible] [constructor] (C : Precategory)
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(H : Π (a b : C) (f : a ⟶ b), is_iso f) : Groupoid :=
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Groupoid.mk C (groupoid.mk C H)
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definition Groupoid.eta [unfold 1] (C : Groupoid) : Groupoid.mk C C = C :=
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Groupoid.rec (λob c, idp) C
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definition Groupoid_of_Group [constructor] (G : Group) : Groupoid :=
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Groupoid.mk unit (groupoid_of_group G)
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end category
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