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getting close in exact couple

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This commit is contained in:
Steve Awodey 2017-05-04 17:46:43 -04:00
parent ed00db374b
commit c67fd11633
1 changed files with 18 additions and 3 deletions
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21
algebra/exact_couple.hlean
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@ -33,7 +33,7 @@ definition homology {B : AbGroup} (d : B →g B) (H : is_differential d) : AbGro
@quotient_ab_group (ab_kernel d) (image_subgroup_of_diff 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))))
@quotient_ab_group (ab_kernel d) (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
@ -50,8 +50,22 @@ definition SES_iso_C {A B C C' : AbGroup} (ses : SES A B C) (k : C ≃g C') : SE
fapply @is_surjective_compose _ _ _ k (SES.g ses),
exact is_surjective_of_is_equiv k,
exact SES.Hg ses,
fapply is_exact.mk,
repeat exact sorry
fapply is_exact.mk,
intro a,
esimp,
note h := SES.ex ses,
note h2 := is_exact.im_in_ker h a,
refine ap k h2 ⬝ _ ,
exact to_respect_one k,
intro b,
intro k3,
note h := SES.ex ses,
note h3 := is_exact.ker_in_im h b,
fapply is_exact.ker_in_im h,
refine _ ⬝ ap k⁻¹ᵍ k3 ⬝ _ ,
esimp,
exact (to_left_inv (equiv_of_isomorphism k) ((SES.g ses) b))⁻¹,
exact to_respect_one k⁻¹ᵍ
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) :=
@ -62,6 +76,7 @@ definition SES_of_differential_ugly {B : AbGroup} (d : B →g B) (H : is_differe
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)),
exact sorry,
end
structure exact_couple (A B : AbGroup) : Type :=