103 lines
3.7 KiB
Text
103 lines
3.7 KiB
Text
/-
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Copyright (c) 2016 Egbert Rijke. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Egbert Rijke, Steve Awodey
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Exact couple, derived couples, and so on
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-/
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import algebra.group_theory hit.set_quotient types.sigma types.list types.sum .quotient_group .subgroup .ses
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open eq algebra is_trunc set_quotient relation sigma sigma.ops prod prod.ops sum list trunc function group trunc
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equiv is_equiv
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definition is_differential {B : AbGroup} (d : B →g B) := Π(b:B), d (d b) = 1
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definition image_subgroup_of_diff {B : AbGroup} (d : B →g B) (H : is_differential d) : subgroup_rel (ab_kernel d) :=
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subgroup_rel_of_subgroup (image_subgroup d) (kernel_subgroup d)
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begin
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intro g p,
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induction p with f, induction f with h p,
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rewrite [p⁻¹],
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esimp,
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exact H h
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end
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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 :=
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begin
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intro b p,
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induction p with q, induction q with b' p, induction p, exact H b'
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end
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definition homology {B : AbGroup} (d : B →g B) (H : is_differential d) : AbGroup :=
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@quotient_ab_group (ab_kernel d) (image_subgroup_of_diff d H)
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definition homology_ugly {B : AbGroup} (d : B →g B) (H : is_differential d) : AbGroup :=
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(quotient_ab_group (image_subgroup (ab_subgroup_of_subgroup_incl (diff_im_in_ker d H))))
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definition homology_iso_ugly {B : AbGroup} (d : B →g B) (H : is_differential d) : (homology d H) ≃g (homology_ugly d H) :=
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begin
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-- fapply quotientgroupiso ...
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exact sorry
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end
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definition SES_iso_C {A B C C' : AbGroup} (ses : SES A B C) (k : C ≃g C') : SES A B C' :=
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begin
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fapply SES.mk,
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exact SES.f ses,
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exact k ∘g SES.g ses,
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exact SES.Hf ses,
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fapply @is_surjective_compose _ _ _ k (SES.g ses),
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exact is_surjective_of_is_equiv k,
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exact SES.Hg ses,
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fapply is_exact.mk,
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repeat exact sorry
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end
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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) :=
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begin
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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)),
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end
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definition SES_of_differential {B : AbGroup} (d : B →g B) (H : is_differential d) : SES (ab_image d) (ab_kernel d) (homology d H) :=
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begin
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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)),
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end
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structure exact_couple (A B : AbGroup) : Type :=
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( i : A →g A) (j : A →g B) (k : B →g A)
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( exact_ij : is_exact i j)
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( exact_jk : is_exact j k)
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( exact_ki : is_exact k i)
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definition differential {A B : AbGroup} (EC : exact_couple A B) : B →g B :=
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(exact_couple.j EC) ∘g (exact_couple.k EC)
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definition differential_is_differential {A B : AbGroup} (EC : exact_couple A B) : is_differential (differential EC) :=
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begin
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induction EC,
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induction exact_jk,
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intro b,
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exact (ap (group_fun j) (im_in_ker (group_fun k b))) ⬝ (respect_one j)
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end
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section derived_couple
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variables {A B : AbGroup} (EC : exact_couple A B)
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definition derived_couple_A : AbGroup :=
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ab_subgroup (image_subgroup (exact_couple.i EC))
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definition derived_couple_B : AbGroup :=
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homology (differential EC) (differential_is_differential EC)
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definition derived_couple_i : derived_couple_A EC →g derived_couple_A EC :=
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(image_lift (exact_couple.i EC)) ∘g (image_incl (exact_couple.i EC))
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definition derived_couple_j : derived_couple_A EC →g derived_couple_B EC :=
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begin
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exact sorry,
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-- refine (comm_gq_map (comm_kernel (boundary CC)) (image_subgroup_of_bd (boundary CC) (boundary_is_boundary CC))) ∘g _,
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end
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end derived_couple
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