lean2/hott/algebra/category/groupoid.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
Ported from Coq HoTT
-/
import .iso ..group
open eq is_trunc iso category algebra nat unit
namespace category
structure groupoid [class] (ob : Type) extends parent : precategory ob :=
mk' :: (all_iso : Π ⦃a b : ob⦄ (f : hom a b), @is_iso ob parent a b f)
abbreviation all_iso := @groupoid.all_iso
attribute groupoid.all_iso [instance] [priority 100000]
definition groupoid.mk [reducible] {ob : Type} (C : precategory ob)
(H : Π (a b : ob) (f : a ⟶ b), is_iso f) : groupoid ob :=
precategory.rec_on C groupoid.mk' H
definition precategory_of_1_type.{l} (A : Type.{l})
[H : is_trunc 1 A] : precategory.{l l} A :=
precategory.mk
(λ (a b : A), a = b)
(λ (a b c : A) (p : b = c) (q : a = b), q ⬝ p)
(λ (a : A), refl a)
(λ (a b c d : A) (p : c = d) (q : b = c) (r : a = b), con.assoc r q p)
(λ (a b : A) (p : a = b), con_idp p)
(λ (a b : A) (p : a = b), idp_con p)
definition groupoid_of_1_type.{l} (A : Type.{l})
[H : is_trunc 1 A] : groupoid.{l l} A :=
groupoid.mk !precategory_of_1_type
(λ (a b : A) (p : a = b), is_iso.mk !con.right_inv !con.left_inv)
-- A groupoid with a contractible carrier is a group
definition group_of_is_contr_groupoid {ob : Type} [H : is_contr ob]
[G : groupoid ob] : group (hom (center ob) (center ob)) :=
begin
fapply group.mk,
intro f g, apply (comp f g),
apply is_hset_hom,
intro f g h, apply (assoc f g h)⁻¹,
apply (ID (center ob)),
intro f, apply id_left,
intro f, apply id_right,
intro f, exact (iso.inverse f),
intro f, exact (iso.left_inverse f),
end
definition group_of_groupoid_unit [G : groupoid unit] : group (hom ⋆ ⋆) :=
begin
fapply group.mk,
intro f g, apply (comp f g),
apply is_hset_hom,
intro f g h, apply (assoc f g h)⁻¹,
apply (ID ⋆),
intro f, apply id_left,
intro f, apply id_right,
intro f, exact (iso.inverse f),
intro f, exact (iso.left_inverse f),
end
-- Conversely we can turn each group into a groupoid on the unit type
definition groupoid_of_group.{l} (A : Type.{l}) [G : group A] : groupoid.{l l} unit :=
begin
fapply groupoid.mk, fapply precategory.mk,
intros, exact A,
intros, apply (@group.is_hset_carrier A G),
intros [a, b, c, g, h], exact (@group.mul A G g h),
intro a, exact (@group.one A G),
intros, exact (@group.mul_assoc A G h g f)⁻¹,
intros, exact (@group.one_mul A G f),
intros, exact (@group.mul_one A G f),
intros, esimp [precategory.mk], apply is_iso.mk,
apply mul.left_inv,
apply mul.right_inv,
end
protected definition hom_group {A : Type} [G : groupoid A] (a : A) :
group (hom a a) :=
begin
fapply group.mk,
intro f g, apply (comp f g),
apply is_hset_hom,
intros f g h, apply (assoc f g h)⁻¹,
apply (ID a),
intro f, apply id_left,
intro f, apply id_right,
intro f, exact (iso.inverse f),
intro f, exact (iso.left_inverse f),
end
-- Bundled version of categories
-- we don't use Groupoid.carrier explicitly, but rather use Groupoid.carrier (to_Precategory C)
structure Groupoid : Type :=
(carrier : Type)
(struct : groupoid carrier)
attribute Groupoid.struct [instance] [coercion]
definition Groupoid.to_Precategory [coercion] [reducible] (C : Groupoid) : Precategory :=
Precategory.mk (Groupoid.carrier C) C
definition groupoid.Mk [reducible] := Groupoid.mk
definition groupoid.MK [reducible] (C : Precategory) (H : Π (a b : C) (f : a ⟶ b), is_iso f)
: Groupoid :=
Groupoid.mk C (groupoid.mk C H)
definition Groupoid.eta (C : Groupoid) : Groupoid.mk C C = C :=
Groupoid.rec (λob c, idp) C
end category