60 lines
2.4 KiB
Text
60 lines
2.4 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
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open path function prod sigma truncation morphism nat path_algebra
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structure foo (A : Type) := (bsp : A)
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structure groupoid [class] (ob : Type) extends precategory ob :=
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(all_iso : Π ⦃a b : ob⦄ (f : hom a b),
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@is_iso ob (precategory.mk hom _ _ _ assoc id_left id_right) a b f)
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namespace groupoid
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instance [persistent] all_iso
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--set_option pp.universes true
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--set_option pp.implicit true
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universe variable l
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definition path_groupoid (A : Type.{l})
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(H : is_trunc 1 A) : groupoid.{l l} A :=
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have C [visible] : precategory.{l l} A, from precategory.mk
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(λ a b, a ≈ b)
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(λ (a b : A), have ish : is_hset (a ≈ b), from succ_is_trunc 0 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), idpath 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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groupoid.mk (precategory.hom)
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(@precategory.homH A C) --(λ (a b : A), have ish : is_hset (a ≈ b), from succ_is_trunc 0 a b, ish)
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(precategory.comp) --(λ (a b c : A) (p : b ≈ c) (q : a ≈ b), q ⬝ p)
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(precategory.ID) --(λ (a : A), idpath a)
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(precategory.assoc) --(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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(precategory.id_left) --(λ (a b : A) (p : a ≈ b), concat_p1 p)
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(precategory.id_right) --(λ (a b : A) (p : a ≈ b), concat_1p p)
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(λ (a b : A) (p : @hom A C a b), @is_iso.mk A C a b p (path.inverse p)
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(have aux : p⁻¹ ⬝ p ≈ idpath b, from concat_Vp p,
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have aux2 : p⁻¹ ∘ p ≈ idpath b, from aux,
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have aux3 : p⁻¹ ∘ p ≈ id, from sorry, aux3)
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(have aux : p ⬝ p⁻¹ ≈ idpath a, from concat_pV p,
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sorry))
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definition group_from_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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end groupoid
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