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This commit is contained in:
Steve Awodey 2017-04-20 16:18:18 -04:00
parent fd5d774e55
commit c313d33b03
1 changed files with 12 additions and 2 deletions
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14
algebra/exact_couple.hlean
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@ -23,15 +23,25 @@ definition image_subgroup_of_diff {B : AbGroup} (d : B →g B) (H : is_different
exact H h
end
definition diff_im_in_ker {B : AbGroup} (d : B →g B) (H : is_differential d) : Π(b : B), image_subgroup d b → kernel_subgroup d b :=
begin
intro b p,
induction p with q, induction q with b' p, induction p, exact H b'
end
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) :=
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),
fapply is_exact.mk,
intro b, induction b,
sorry,
-- use the more general fact that a subgroup inclusion is a group homomorphism
-- maybe use SES_of_subgroup?
end
structure exact_couple (A B : AbGroup) : Type :=