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Merge branch 'master' of https://github.com/cmu-phil/Spectral

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Egbert Rijke 2017-04-27 18:08:25 -04:00
commit 454401fdea
2 changed files with 36 additions and 25 deletions
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34
algebra/exact_couple.hlean
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@ -32,16 +32,36 @@ definition diff_im_in_ker {B : AbGroup} (d : B →g B) (H : is_differential d) :
definition homology {B : AbGroup} (d : B →g B) (H : is_differential d) : AbGroup :=
@quotient_ab_group (ab_kernel d) (image_subgroup_of_diff d H)
definition SES_of_differential {B : AbGroup} (d : B →g B) (H : is_differential d) : SES (ab_image d) (ab_kernel d) (homology d H) :=
definition homology_ugly {B : AbGroup} (d : B →g B) (H : is_differential d) : AbGroup :=
(quotient_ab_group (image_subgroup (ab_subgroup_of_subgroup_incl (diff_im_in_ker d H))))
definition homology_iso_ugly {B : AbGroup} (d : B →g B) (H : is_differential d) : (homology d H) ≃g (homology_ugly d H) :=
begin
-- fapply quotientgroupiso ...
exact sorry
end
definition SES_iso_C {A B C C' : AbGroup} (ses : SES A B C) (k : C ≃g C') : SES A B C' :=
begin
fapply SES.mk,
exact @ab_subgroup_of_subgroup_incl B (image_subgroup d) (kernel_subgroup d) (diff_im_in_ker d H),
exact ab_qg_map (image_subgroup_of_diff d H),
rexact is_embedding_ab_subgroup_of_subgroup_incl (diff_im_in_ker d H),
exact is_surjective_ab_qg_map (image_subgroup_of_diff d H),
exact SES.f ses,
exact k ∘g SES.g ses,
exact SES.Hf ses,
fapply @is_surjective_compose _ _ _ k (SES.g ses),
exact is_surjective_of_is_equiv k,
exact SES.Hg ses,
fapply is_exact.mk,
intro b, induction b,
sorry,
repeat exact sorry
end
definition SES_of_differential_ugly {B : AbGroup} (d : B →g B) (H : is_differential d) : SES (ab_image d) (ab_kernel d) (homology_ugly d H) :=
begin
exact SES_of_inclusion (ab_subgroup_of_subgroup_incl (diff_im_in_ker d H)) (is_embedding_ab_subgroup_of_subgroup_incl (diff_im_in_ker d H)),
end
definition SES_of_differential {B : AbGroup} (d : B →g B) (H : is_differential d) : SES (ab_image d) (ab_kernel d) (homology d H) :=
begin
exact SES_of_inclusion (ab_subgroup_of_subgroup_incl (diff_im_in_ker d H)) (is_embedding_ab_subgroup_of_subgroup_incl (diff_im_in_ker d H)),
end
structure exact_couple (A B : AbGroup) : Type :=

23
algebra/subgroup.hlean
View file

@ -284,24 +284,15 @@ namespace group
exact H
end
definition ab_image {G : AbGroup} {H : Group} (f : G →g H) : AbGroup :=
AbGroup_of_Group (image f)
begin
intro g h,
induction g with x t, induction h with y s,
fapply subtype_eq,
induction t with p, induction s with q, induction p with g p, induction q with h q, induction p, induction q,
refine (((respect_mul f g h)⁻¹ ⬝ _) ⬝ (respect_mul f h g)),
apply (ap f),
induction G, induction struct, apply mul_comm
end
definition ab_image {G : AbGroup} {H : AbGroup} (f : G →g H) : AbGroup :=
ab_subgroup (image_subgroup f)
definition image_incl {G H : Group} (f : G →g H) : image f →g H :=
definition image_incl {G H : Group} (f : G →g H) : image f →g H :=
incl_of_subgroup (image_subgroup f)
definition ab_image_incl {A B : AbGroup} (f : A →g B) : ab_image f →g B := incl_of_subgroup (image_subgroup f)
definition ab_image_incl {A B : AbGroup} (f : A →g B) : ab_image f →g B := incl_of_subgroup (image_subgroup f)
definition is_equiv_surjection_ab_image_incl {A B : AbGroup} (f : A →g B) (H : is_surjective f) : is_equiv (ab_image_incl f ) :=
definition is_equiv_surjection_ab_image_incl {A B : AbGroup} (f : A →g B) (H : is_surjective f) : is_equiv (ab_image_incl f ) :=
begin
fapply is_equiv.adjointify (ab_image_incl f),
intro b,
@ -315,14 +306,14 @@ namespace group
reflexivity
end
definition iso_surjection_ab_image_incl [constructor] {A B : AbGroup} (f : A →g B) (H : is_surjective f) : ab_image f ≃g B :=
definition iso_surjection_ab_image_incl [constructor] {A B : AbGroup} (f : A →g B) (H : is_surjective f) : ab_image f ≃g B :=
begin
fapply isomorphism.mk,
exact (ab_image_incl f),
exact is_equiv_surjection_ab_image_incl f H
end
definition hom_lift [constructor] {G H : Group} (f : G →g H) (K : subgroup_rel H) (Hyp : Π (g : G), K (f g)) : G →g subgroup K :=
definition hom_lift [constructor] {G H : Group} (f : G →g H) (K : subgroup_rel H) (Hyp : Π (g : G), K (f g)) : G →g subgroup K :=
begin
fapply homomorphism.mk,
intro g,