92 lines
2.9 KiB
Text
92 lines
2.9 KiB
Text
-- 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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-- Ported from Coq HoTT
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import .precategory.basic .precategory.morphism .group types.pi
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open eq function prod sigma pi truncation morphism nat path_algebra unit prod sigma.ops
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structure foo (A : Type) := (bsp : A)
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structure groupoid [class] (ob : Type) extends parent : precategory ob :=
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(all_iso : Π ⦃a b : ob⦄ (f : hom a b),
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@is_iso ob parent a b f)
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namespace groupoid
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attribute all_iso [instance]
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universe variable l
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open precategory
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definition path_groupoid (A : Type.{l})
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(H : is_trunc (nat.zero .+1) A) : groupoid.{l l} A :=
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groupoid.mk
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(λ (a b : A), a = b)
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(λ (a b : A), have ish : is_hset (a = b), from succ_is_trunc nat.zero a b, ish)
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(λ (a b c : A) (p : b = c) (q : a = b), q ⬝ p)
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(λ (a : A), refl a)
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(λ (a b c d : A) (p : c = d) (q : b = c) (r : a = b), concat_pp_p r q p)
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(λ (a b : A) (p : a = b), concat_p1 p)
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(λ (a b : A) (p : a = b), concat_1p p)
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(λ (a b : A) (p : a = b), @is_iso.mk A _ a b p (p⁻¹)
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!concat_pV !concat_Vp)
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-- A groupoid with a contractible carrier is a group
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definition group_of_contr {ob : Type} (H : is_contr ob)
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(G : groupoid ob) : group (hom (center ob) (center ob)) :=
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begin
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fapply group.mk,
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intros (f, g), apply (comp f g),
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apply homH,
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intros (f, g, h), apply ((assoc f g h)⁻¹),
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apply (ID (center ob)),
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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 (morphism.inverse f),
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intro f, exact (morphism.inverse_compose f),
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end
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definition group_of_unit (G : groupoid unit) : group (hom ⋆ ⋆) :=
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begin
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fapply group.mk,
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intros (f, g), apply (comp f g),
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apply homH,
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intros (f, g, h), apply ((assoc f g h)⁻¹),
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apply (ID ⋆),
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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 (morphism.inverse f),
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intro f, exact (morphism.inverse_compose f),
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end
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-- Conversely we can turn each group into a groupoid on the unit type
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definition of_group (A : Type.{l}) [G : group A] : groupoid.{l l} unit :=
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begin
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fapply groupoid.mk,
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intros, exact A,
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intros, apply (@group.carrier_hset A G),
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intros (a, b, c, g, h), exact (@group.mul A G g h),
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intro a, exact (@group.one A G),
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intros, exact ((@group.mul_assoc A G h g f)⁻¹),
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intros, exact (@group.mul_left_id A G f),
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intros, exact (@group.mul_right_id A G f),
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intros, apply is_iso.mk,
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apply mul_left_inv,
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apply mul_right_inv,
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end
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protected definition hom_group {A : Type} [G : groupoid A] (a : A) :
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group (hom a a) :=
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begin
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fapply group.mk,
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intros (f, g), apply (comp f g),
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apply homH,
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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 (morphism.inverse f),
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intro f, exact (morphism.inverse_compose f),
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end
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end groupoid
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