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feat(library/hott) prove that each group is a contractible groupoid

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Jakob von Raumer 2014-12-09 15:00:30 -05:00
parent f023e4999c
commit 86a38c6c3d
1 changed files with 32 additions and 5 deletions
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library/hott/algebra/groupoid.lean
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@ -4,7 +4,7 @@
-- Ported from Coq HoTT -- Ported from Coq HoTT
import .precategory.basic .precategory.morphism .group import .precategory.basic .precategory.morphism .group
open path function prod sigma truncation morphism nat path_algebra open path function prod sigma truncation morphism nat path_algebra unit
structure foo (A : Type) := (bsp : A) structure foo (A : Type) := (bsp : A)
@ -20,8 +20,18 @@ instance [persistent] all_iso
--set_option pp.implicit true --set_option pp.implicit true
universe variable l universe variable l
definition path_groupoid (A : Type.{l}) definition path_groupoid (A : Type.{l})
(H : is_trunc 1 A) : groupoid.{l l} A := (H : is_trunc (nat.zero .+1) A) : groupoid.{l l} A :=
have C [visible] : precategory.{l l} A, from precategory.mk groupoid.mk
(λ (a b : A), a ≈ b)
(λ (a b : A), have ish : is_hset (a ≈ b), from succ_is_trunc nat.zero a b, ish)
(λ (a b c : A) (p : b ≈ c) (q : a ≈ b), q ⬝ p)
(λ (a : A), idpath a)
(λ (a b c d : A) (p : c ≈ d) (q : b ≈ c) (r : a ≈ b), concat_pp_p r q p)
(λ (a b : A) (p : a ≈ b), concat_p1 p)
(λ (a b : A) (p : a ≈ b), concat_1p p)
(λ (a b : A) (p : a ≈ b), @is_iso.mk A _ a b p (path.inverse p)
sorry sorry)
/-have C [visible] : precategory.{l l} A, from precategory.mk
(λ a b, a ≈ b) (λ a b, a ≈ b)
(λ (a b : A), have ish : is_hset (a ≈ b), from succ_is_trunc 0 a b, ish) (λ (a b : A), have ish : is_hset (a ≈ b), from succ_is_trunc 0 a b, ish)
(λ (a b c : A) (p : b ≈ c) (q : a ≈ b), q ⬝ p) (λ (a b c : A) (p : b ≈ c) (q : a ≈ b), q ⬝ p)
@ -41,9 +51,10 @@ groupoid.mk (precategory.hom)
have aux2 : p⁻¹ ∘ p ≈ idpath b, from aux, have aux2 : p⁻¹ ∘ p ≈ idpath b, from aux,
have aux3 : p⁻¹ ∘ p ≈ id, from sorry, aux3) have aux3 : p⁻¹ ∘ p ≈ id, from sorry, aux3)
(have aux : p ⬝ p⁻¹ ≈ idpath a, from concat_pV p, (have aux : p ⬝ p⁻¹ ≈ idpath a, from concat_pV p,
sorry)) sorry))-/
definition group_from_contr {ob : Type} (H : is_contr ob) -- A groupoid with a contractible carrier is a group
definition group_of_contr {ob : Type} (H : is_contr ob)
(G : groupoid ob) : group (hom (center ob) (center ob)) := (G : groupoid ob) : group (hom (center ob) (center ob)) :=
begin begin
fapply group.mk, fapply group.mk,
@ -57,4 +68,20 @@ begin
intro f, exact (morphism.inverse_compose f), intro f, exact (morphism.inverse_compose f),
end end
-- Conversely we can turn each group into a groupoid on the unit type
definition of_group {A : Type.{l}} (G : group A) : groupoid.{l l} unit :=
begin
fapply groupoid.mk,
intros, exact A,
intros, apply (@group.carrier_hset 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.mul_left_id A G f),
intros, exact (@group.mul_right_id A G f),
intros, apply is_iso.mk,
apply mul_left_inv,
apply mul_right_inv,
end
end groupoid end groupoid