Spectral/algebra/exact_couple.hlean

/-
Copyright (c) 2016 Egbert Rijke. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Egbert Rijke

Exact couple, derived couples, and so on
-/

import algebra.group_theory hit.set_quotient types.sigma types.list types.sum .quotient_group .subgroup

open eq algebra is_trunc set_quotient relation sigma sigma.ops prod prod.ops sum list trunc function group trunc
     equiv

structure is_exact {A B C : AbGroup} (f : A →g B) (g : B →g C) :=
  ( im_in_ker : Π(a:A), g (f a) = 1)
  ( ker_in_im : Π(b:B), (g b = 1) → image_subgroup f b)

structure SES (A B C : AbGroup) :=
  ( f : A →g B)
  ( g : B →g C)
  ( Hf : is_embedding f)
  ( Hg : is_surjective g)
  ( ex : is_exact f g)

structure hom_SES {A B C A' B' C' : AbGroup} (ses : SES A B C) (ses' : SES A' B' C') :=
  ( hA : A →g A')
  ( hB : B →g B')
  ( hC : C →g C')
  ( htpy1 : hB ∘g (SES.f ses) ~ (SES.f ses') ∘g hA)
  ( htpy2 : hC ∘g (SES.g ses) ~ (SES.g ses') ∘g hB)

--definition quotient_SES {A B C : AbGroup} (ses : SES A B C) : 
--  quotient_ab_group (image_subgroup (SES.f ses)) ≃g C :=
--  begin
--    fapply ab_group_first_iso_thm B C (SES.g ses), 
--  end

-- definition pre_right_extend_SES (to separate the following definition and replace C with B/A)

definition right_extend_SES {A B C A' B' C' : AbGroup} 
  (ses : SES A B C) (ses' : SES A' B' C')  
  (hA : A →g A') (hB : B →g B') (htpy1 : hB ∘g (SES.f ses) ~ (SES.f ses') ∘g hA) : C →g C' :=
  begin
    refine _ ∘g (codomain_surjection_is_quotient (SES.g ses) (SES.Hg ses))⁻¹ᵍ,
    refine (codomain_surjection_is_quotient (SES.g ses') (SES.Hg ses')) ∘g _,
    fapply ab_group_quotient_homomorphism B B' (kernel_subgroup (SES.g ses)) (kernel_subgroup (SES.g ses')) hB,
    intro b,
    intro K,
    have k : trunctype.carrier (image_subgroup (SES.f ses) b), from is_exact.ker_in_im (SES.ex ses) b K,
    induction k, induction a with a p,
    rewrite [p⁻¹],
    rewrite [htpy1 a],
    fapply is_exact.im_in_ker (SES.ex ses') (hA a), 
  end

definition right_extend_hom_SES {A B C A' B' C' : AbGroup}
  (ses : SES A B C) (ses' : SES A' B' C')
  (hA : A →g A') (hB : B →g B') (htpy1 : hB ∘g (SES.f ses) ~ (SES.f ses') ∘g hA) : hom_SES ses ses' :=
  begin
    fapply hom_SES.mk,
    exact hA,
    exact hB,
    exact right_extend_SES ses ses' hA hB htpy1,
    exact htpy1,
    exact sorry -- fapply quotient_group_compute,
  end

definition is_differential {B : AbGroup} (d : B →g B) := Π(b:B), d (d b) = 1

definition image_subgroup_of_diff {B : AbGroup} (d : B →g B) (H : is_differential d) : subgroup_rel (ab_kernel d) :=
  subgroup_rel_of_subgroup (image_subgroup d) (kernel_subgroup d)
  begin
    intro g p,
    induction p with f, induction f with h p,
    rewrite [p⁻¹],
    esimp,
    exact H h
  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)

structure exact_couple (A B : AbGroup) : Type :=
  ( i : A →g A) (j : A →g B) (k : B →g A)
  ( exact_ij : is_exact i j)
  ( exact_jk : is_exact j k)
  ( exact_ki : is_exact k i)

definition differential {A B : AbGroup} (EC : exact_couple A B) : B →g B :=
  (exact_couple.j EC) ∘g (exact_couple.k EC)

definition differential_is_differential {A B : AbGroup} (EC : exact_couple A B) : is_differential (differential EC) :=
  begin
    induction EC,
    induction exact_jk,
    intro b,
    exact (ap (group_fun j) (im_in_ker (group_fun k b))) ⬝ (respect_one j)
  end

section derived_couple

  variables {A B : AbGroup} (EC : exact_couple A B)

definition derived_couple_A : AbGroup :=
  ab_subgroup (image_subgroup (exact_couple.i EC))

definition derived_couple_B : AbGroup :=
  homology (differential EC) (differential_is_differential EC)

definition derived_couple_i : derived_couple_A EC →g derived_couple_A EC :=
  (image_lift (exact_couple.i EC)) ∘g (image_incl (exact_couple.i EC))

definition derived_couple_j : derived_couple_A EC →g derived_couple_B EC :=
  begin
  exact sorry,
--    refine (comm_gq_map (comm_kernel (boundary CC)) (image_subgroup_of_bd (boundary CC) (boundary_is_boundary CC))) ∘g _,
  end

end derived_couple