wip
This commit is contained in:
parent
0b1d4428b3
commit
922baa9975
3 changed files with 57 additions and 8 deletions
|
@ -1 +0,0 @@
|
||||||
Steve@steveawodeysAir.wv.cc.cmu.edu.268
|
|
|
@ -35,12 +35,20 @@ definition homology {B : AbGroup} (d : B →g B) (H : is_differential d) : AbGro
|
||||||
definition homology_ugly {B : AbGroup} (d : B →g B) (H : is_differential d) : AbGroup :=
|
definition homology_ugly {B : AbGroup} (d : B →g B) (H : is_differential d) : AbGroup :=
|
||||||
@quotient_ab_group (ab_kernel d) (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) :=
|
definition homology_iso_ugly {B : AbGroup} (d : B →g B) (H : is_differential d) : (homology d H) ≃g (homology_ugly d H) :=
|
||||||
begin
|
begin
|
||||||
fapply @iso_of_ab_qg_group (ab_kernel d) (image_subgroup (ab_subgroup_of_subgroup_incl (diff_im_in_ker d H))) (image_subgroup_of_diff d H),
|
fapply @iso_of_ab_qg_group (ab_kernel d),
|
||||||
|
intro a,
|
||||||
|
intro p, induction p with f, induction f with b p,
|
||||||
|
fapply tr, fapply fiber.mk, fapply sigma.mk, exact d b, fapply tr, fapply fiber.mk, exact b, reflexivity,
|
||||||
|
induction a with c q, fapply subtype_eq, refine p ⬝ _, reflexivity,
|
||||||
|
intro b p, induction p with f, induction f with c p, induction p,
|
||||||
|
induction c with a q, induction q with f, induction f with a' p, induction p,
|
||||||
|
fapply tr, fapply fiber.mk, exact a', reflexivity
|
||||||
end
|
end
|
||||||
|
|
||||||
|
|
||||||
definition SES_iso_C {A B C C' : AbGroup} (ses : SES A B C) (k : C ≃g C') : SES A B C' :=
|
definition SES_iso_C {A B C C' : AbGroup} (ses : SES A B C) (k : C ≃g C') : SES A B C' :=
|
||||||
begin
|
begin
|
||||||
fapply SES.mk,
|
fapply SES.mk,
|
||||||
|
@ -68,6 +76,7 @@ definition SES_iso_C {A B C C' : AbGroup} (ses : SES A B C) (k : C ≃g C') : SE
|
||||||
exact to_respect_one k⁻¹ᵍ
|
exact to_respect_one k⁻¹ᵍ
|
||||||
end
|
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) :=
|
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
|
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 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)),
|
||||||
|
@ -75,8 +84,9 @@ 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) :=
|
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
|
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)),
|
fapply SES_iso_C,
|
||||||
exact sorry,
|
fapply 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 (homology_iso_ugly d H)⁻¹ᵍ
|
||||||
end
|
end
|
||||||
|
|
||||||
structure exact_couple (A B : AbGroup) : Type :=
|
structure exact_couple (A B : AbGroup) : Type :=
|
||||||
|
@ -98,17 +108,38 @@ definition differential_is_differential {A B : AbGroup} (EC : exact_couple A B)
|
||||||
|
|
||||||
section derived_couple
|
section derived_couple
|
||||||
|
|
||||||
variables {A B : AbGroup} (EC : exact_couple A B)
|
parameters {A B : AbGroup} (EC : exact_couple A B)
|
||||||
|
local abbreviation i := exact_couple.i EC
|
||||||
|
local abbreviation j := exact_couple.j EC
|
||||||
|
local abbreviation k := exact_couple.k EC
|
||||||
|
local abbreviation d := differential EC
|
||||||
|
local abbreviation H := differential_is_differential EC
|
||||||
|
|
||||||
definition derived_couple_A : AbGroup :=
|
definition derived_couple_A : AbGroup :=
|
||||||
ab_subgroup (image_subgroup (exact_couple.i EC))
|
ab_subgroup (image_subgroup i)
|
||||||
|
|
||||||
definition derived_couple_B : AbGroup :=
|
definition derived_couple_B : AbGroup :=
|
||||||
homology (differential EC) (differential_is_differential EC)
|
homology (differential EC) (differential_is_differential EC)
|
||||||
|
|
||||||
definition derived_couple_i : derived_couple_A EC →g derived_couple_A EC :=
|
definition derived_couple_i : derived_couple_A →g derived_couple_A :=
|
||||||
(image_lift (exact_couple.i EC)) ∘g (image_incl (exact_couple.i EC))
|
(image_lift (exact_couple.i EC)) ∘g (image_incl (exact_couple.i EC))
|
||||||
|
|
||||||
|
definition SES_of_exact_couple_at_i : SES (ab_kernel i) A (ab_image i) :=
|
||||||
|
begin
|
||||||
|
fapply SES_iso_C,
|
||||||
|
fapply SES_of_subgroup (kernel_subgroup i),
|
||||||
|
fapply ab_group_first_iso_thm i,
|
||||||
|
end
|
||||||
|
|
||||||
|
definition j_factor : A →g (ab_kernel d) :=
|
||||||
|
begin
|
||||||
|
fapply ab_hom_lift j,
|
||||||
|
intro a,
|
||||||
|
unfold kernel_subgroup,
|
||||||
|
exact sorry --refine ap (group_fun j) _,
|
||||||
|
end
|
||||||
|
|
||||||
|
|
||||||
definition derived_couple_j : derived_couple_A EC →g derived_couple_B EC :=
|
definition derived_couple_j : derived_couple_A EC →g derived_couple_B EC :=
|
||||||
begin
|
begin
|
||||||
exact sorry,
|
exact sorry,
|
||||||
|
|
|
@ -319,6 +319,16 @@ definition hom_lift [constructor] {G H : Group} (f : G →g H) (K : subgroup_rel
|
||||||
intro g h, apply subtype_eq, esimp, apply respect_mul
|
intro g h, apply subtype_eq, esimp, apply respect_mul
|
||||||
end
|
end
|
||||||
|
|
||||||
|
definition ab_hom_lift [constructor] {G H : AbGroup} (f : G →g H) (K : subgroup_rel H) (Hyp : Π (g : G), K (f g)) : G →g ab_subgroup K :=
|
||||||
|
begin
|
||||||
|
fapply homomorphism.mk,
|
||||||
|
intro g,
|
||||||
|
fapply sigma.mk,
|
||||||
|
exact f g,
|
||||||
|
fapply Hyp,
|
||||||
|
intro g h, apply subtype_eq, apply respect_mul,
|
||||||
|
end
|
||||||
|
|
||||||
definition image_lift [constructor] {G H : Group} (f : G →g H) : G →g image f :=
|
definition image_lift [constructor] {G H : Group} (f : G →g H) : G →g image f :=
|
||||||
begin
|
begin
|
||||||
fapply hom_lift f,
|
fapply hom_lift f,
|
||||||
|
@ -328,6 +338,15 @@ definition hom_lift [constructor] {G H : Group} (f : G →g H) (K : subgroup_rel
|
||||||
exact g, reflexivity
|
exact g, reflexivity
|
||||||
end
|
end
|
||||||
|
|
||||||
|
definition ab_image_lift [constructor] {G H : AbGroup} (f : G →g H) : G →g image f :=
|
||||||
|
begin
|
||||||
|
fapply hom_lift f,
|
||||||
|
intro g,
|
||||||
|
apply tr,
|
||||||
|
fapply fiber.mk,
|
||||||
|
exact g, reflexivity
|
||||||
|
end
|
||||||
|
|
||||||
definition is_surjective_image_lift {G H : Group} (f : G →g H) : is_surjective (image_lift f) :=
|
definition is_surjective_image_lift {G H : Group} (f : G →g H) : is_surjective (image_lift f) :=
|
||||||
begin
|
begin
|
||||||
intro h,
|
intro h,
|
||||||
|
|
Loading…
Add table
Reference in a new issue