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WIP on exact couple and basic group theory

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This commit is contained in:
Steve Awodey 2016-11-10 16:49:09 -05:00
parent e4c5d34d10
commit e87c235e24
3 changed files with 19 additions and 17 deletions
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1
algebra/.#exact_couple.hlean
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@ -1 +0,0 @@
Steve@steveawodeysAir.wv.cc.cmu.edu.255

30
algebra/exact_couple.hlean
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@ -15,9 +15,9 @@ structure is_exact {A B C : CommGroup} (f : A →g B) (g : B →g C) :=
( im_in_ker : Π(a:A), g (f a) = 1) ( im_in_ker : Π(a:A), g (f a) = 1)
( ker_in_im : Π(b:B), (g b = 1) → ∃(a:A), f a = b) ( ker_in_im : Π(b:B), (g b = 1) → ∃(a:A), f a = b)
definition is_boundary {B : CommGroup} (d : B →g B) := Π(b:B), d (d b) = 1 definition is_differential {B : CommGroup} (d : B →g B) := Π(b:B), d (d b) = 1
definition image_subgroup_of_bd {B : CommGroup} (d : B →g B) (H : is_boundary d) : subgroup_rel (comm_kernel d) := definition image_subgroup_of_diff {B : CommGroup} (d : B →g B) (H : is_differential d) : subgroup_rel (comm_kernel d) :=
subgroup_rel_of_subgroup (image_subgroup d) (kernel_subgroup d) subgroup_rel_of_subgroup (image_subgroup d) (kernel_subgroup d)
begin begin
intro g p, intro g p,
@ -27,8 +27,8 @@ definition image_subgroup_of_bd {B : CommGroup} (d : B →g B) (H : is_boundary
exact H h exact H h
end end
definition homology {B : CommGroup} (d : B →g B) (H : is_boundary d) : CommGroup := definition homology {B : CommGroup} (d : B →g B) (H : is_differential d) : CommGroup :=
@quotient_comm_group (comm_kernel d) (image_subgroup_of_bd d H) @quotient_comm_group (comm_kernel d) (image_subgroup_of_diff d H)
structure exact_couple (A B : CommGroup) : Type := structure exact_couple (A B : CommGroup) : Type :=
( i : A →g A) (j : A →g B) (k : B →g A) ( i : A →g A) (j : A →g B) (k : B →g A)
@ -36,12 +36,12 @@ structure exact_couple (A B : CommGroup) : Type :=
( exact_jk : is_exact j k) ( exact_jk : is_exact j k)
( exact_ki : is_exact k i) ( exact_ki : is_exact k i)
definition boundary {A B : CommGroup} (CC : exact_couple A B) : B →g B := definition differential {A B : CommGroup} (EC : exact_couple A B) : B →g B :=
(exact_couple.j CC) ∘g (exact_couple.k CC) (exact_couple.j EC) ∘g (exact_couple.k EC)
definition boundary_is_boundary {A B : CommGroup} (CC : exact_couple A B) : is_boundary (boundary CC) := definition differential_is_differential {A B : CommGroup} (EC : exact_couple A B) : is_differential (differential EC) :=
begin begin
induction CC, induction EC,
induction exact_jk, induction exact_jk,
intro b, intro b,
exact (ap (group_fun j) (im_in_ker (group_fun k b))) ⬝ (respect_one j) exact (ap (group_fun j) (im_in_ker (group_fun k b))) ⬝ (respect_one j)
@ -49,21 +49,21 @@ definition boundary_is_boundary {A B : CommGroup} (CC : exact_couple A B) : is_b
section derived_couple section derived_couple
variables {A B : CommGroup} (CC : exact_couple A B) variables {A B : CommGroup} (EC : exact_couple A B)
definition derived_couple_A : CommGroup := definition derived_couple_A : CommGroup :=
comm_subgroup (image_subgroup (exact_couple.i CC)) comm_subgroup (image_subgroup (exact_couple.i EC))
definition derived_couple_B : CommGroup := definition derived_couple_B : CommGroup :=
homology (boundary CC) (boundary_is_boundary CC) homology (differential EC) (differential_is_differential EC)
definition derived_couple_i : derived_couple_A CC →g derived_couple_A CC := definition derived_couple_i : derived_couple_A EC →g derived_couple_A EC :=
(image_lift (exact_couple.i CC)) ∘g (image_incl (exact_couple.i CC)) (image_lift (exact_couple.i EC)) ∘g (image_incl (exact_couple.i EC))
definition derived_couple_j : derived_couple_A CC →g derived_couple_B CC := definition derived_couple_j : derived_couple_A EC →g derived_couple_B EC :=
begin begin
exact sorry,
-- refine (comm_gq_map (comm_kernel (boundary CC)) (image_subgroup_of_bd (boundary CC) (boundary_is_boundary CC))) ∘g _, -- refine (comm_gq_map (comm_kernel (boundary CC)) (image_subgroup_of_bd (boundary CC) (boundary_is_boundary CC))) ∘g _,
exact sorry
end end
end derived_couple end derived_couple

5
algebra/quotient_group.hlean
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@ -248,7 +248,10 @@ definition comm_group_first_iso_thm (A B : CommGroup) (f : A →g B) : quotient_
fapply image_lift, fapply image_lift,
intro a, intro a,
intro k, intro k,
fapply comm_gq_map_eq_one (f a), fapply image_incl_eq_one,
exact k,
exact sorry,
-- show that the above map is injective and surjective.
end end
/- set generating normal subgroup -/ /- set generating normal subgroup -/