quotient_extend_unique_SES
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Egbert Rijke
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2 changed files with 117 additions and 25 deletions
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@ -243,18 +243,31 @@ namespace group
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exact K g
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end
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definition ab_gelim_unique {K : subgroup_rel A} (f : A →g B) (H : Π (a :A), K a → f a = 1) (k : quotient_ab_group K →g B)
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: ( k ∘g ab_qg_map K ~ f) → k ~ quotient_group_elim f H :=
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begin
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fapply gelim_unique,
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end
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definition qg_universal_property (f : G →g G') (H : Π⦃g⦄, N g → f g = 1) :
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is_contr (Σ(g : quotient_group N →g G'), g ∘g qg_map N = f) :=
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is_contr (Σ(g : quotient_group N →g G'), g ∘g qg_map N ~ f) :=
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begin
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fapply is_contr.mk,
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-- give center of contraction
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{ fapply sigma.mk, exact quotient_group_elim f H, apply homomorphism_eq, exact quotient_group_compute f H },
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{ fapply sigma.mk, exact quotient_group_elim f H, exact quotient_group_compute f H },
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-- give contraction
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{ intro pair, induction pair with g p, fapply sigma_eq,
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{esimp, apply homomorphism_eq, symmetry, exact gelim_unique f H g (homotopy_of_homomorphism_eq p)},
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{esimp, apply homomorphism_eq, symmetry, exact gelim_unique f H g p},
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{fapply is_prop.elimo} }
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end
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definition ab_qg_universal_property {K : subgroup_rel A} (f : A →g B) (H : Π (a :A), K a → f a = 1) :
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is_contr ((Σ(g : quotient_ab_group K →g B), g ∘g ab_qg_map K ~ f) ) :=
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begin
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fapply qg_universal_property,
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exact H
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end
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------------------------------------------------
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-- FIRST ISOMORPHISM THEOREM
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------------------------------------------------
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@ -3,7 +3,9 @@ Copyright (c) 2017 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
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Exact couple, derived couples, and so on
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Basic facts about short exact sequences.
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At the moment, it only covers short exact sequences of abelian groups, but this should be extended to short exact sequences in any abelian category.
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-/
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import algebra.group_theory hit.set_quotient types.sigma types.list types.sum .quotient_group .subgroup
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@ -38,6 +40,18 @@ definition SES_of_inclusion {A B : AbGroup} (f : A →g B) (Hf : is_embedding f)
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fapply rel_of_ab_qg_map_eq_one, assumption
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end
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definition SES_of_subgroup {B : AbGroup} (S : subgroup_rel B) : SES (ab_subgroup S) B (quotient_ab_group S) :=
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begin
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fapply SES.mk,
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exact incl_of_subgroup S,
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exact ab_qg_map S,
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exact is_embedding_incl_of_subgroup S,
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exact is_surjective_ab_qg_map S,
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fapply is_exact.mk,
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intro a, fapply ab_qg_map_eq_one, induction a with b p, exact p,
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intro b p, fapply tr, fapply fiber.mk, fapply sigma.mk b, fapply rel_of_ab_qg_map_eq_one, exact p, reflexivity,
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end
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definition SES_of_surjective_map {B C : AbGroup} (g : B →g C) (Hg : is_surjective g) : SES (ab_kernel g) B C :=
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begin
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fapply SES.mk,
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@ -57,6 +71,15 @@ structure hom_SES {A B C A' B' C' : AbGroup} (ses : SES A B C) (ses' : SES A' B'
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( htpy1 : hB ∘g (SES.f ses) ~ (SES.f ses') ∘g hA)
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( htpy2 : hC ∘g (SES.g ses) ~ (SES.g ses') ∘g hB)
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section ses
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parameters {A B C : AbGroup} (ses : SES A B C)
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local abbreviation f := SES.f ses
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local notation `g` := SES.g ses
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local abbreviation ex := SES.ex ses
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local abbreviation q := ab_qg_map (kernel_subgroup g)
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local abbreviation B_mod_A := quotient_ab_group (kernel_subgroup g)
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--definition quotient_SES {A B C : AbGroup} (ses : SES A B C) :
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-- quotient_ab_group (image_subgroup (SES.f ses)) ≃g C :=
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-- begin
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@ -65,34 +88,63 @@ structure hom_SES {A B C A' B' C' : AbGroup} (ses : SES A B C) (ses' : SES A' B'
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-- definition pre_right_extend_SES (to separate the following definition and replace C with B/A)
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definition quotient_codomain_SES {A B C : AbGroup} (ses : SES A B C) : quotient_ab_group (kernel_subgroup (SES.g ses)) ≃g C :=
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definition quotient_codomain_SES : B_mod_A ≃g C :=
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begin
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exact (codomain_surjection_is_quotient (SES.g ses) (SES.Hg ses))
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exact (codomain_surjection_is_quotient g (SES.Hg ses))
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end
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definition quotient_triangle_SES {A B C : AbGroup} (ses : SES A B C) : (quotient_codomain_SES ses) ∘g (ab_qg_map (kernel_subgroup (SES.g ses))) ~ (SES.g ses) :=
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local abbreviation α := quotient_codomain_SES
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definition quotient_triangle_SES : α ∘g q ~ g :=
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begin
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reflexivity
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end
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section short_exact_sequences
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parameters {A B C A' B' C' : AbGroup}
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(ses : SES A B C) (ses' : SES A' B' C')
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(hA : A →g A') (hB : B →g B') (htpy1 : hB ∘g (SES.f ses) ~ (SES.f ses') ∘g hA)
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definition quotient_triangle_extend_SES {C': AbGroup} (k : B →g C') :
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(Σ (h : C →g C'), h ∘g g ~ k) ≃ (Σ (h' : B_mod_A →g C'), h' ∘g q ~ k) :=
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begin
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fapply equiv.mk,
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intro pair, induction pair with h H,
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fapply sigma.mk, exact h ∘g α, intro b,
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exact H b,
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fapply adjointify,
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intro pair, induction pair with h' H', fapply sigma.mk,
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exact h' ∘g α⁻¹ᵍ,
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intro b,
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exact calc
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h' (α⁻¹ᵍ (g b)) = h' (α⁻¹ᵍ (α (q b))) : by reflexivity
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... = h' (q b) : by exact hwhisker_left h' (left_inv α) (q b)
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... = k b : by exact H' b,
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intro pair, induction pair with h' H', fapply sigma_eq, esimp, fapply homomorphism_eq, fapply hwhisker_left h' (left_inv α), esimp, fapply is_prop.elimo, fapply pi.is_trunc_pi, intro a, fapply is_trunc_eq,
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intro pair, induction pair with h H, fapply sigma_eq, esimp, fapply homomorphism_eq, fapply hwhisker_left h (right_inv α),
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esimp, fapply is_prop.elimo, fapply pi.is_trunc_pi, intro a, fapply is_trunc_eq,
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end
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parameters {A' B' C' : AbGroup}
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(ses' : SES A' B' C')
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(hA : A →g A') (hB : B →g B') (htpy1 : hB ∘g f ~ (SES.f ses') ∘g hA)
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local abbreviation f := SES.f ses
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local abbreviation g := SES.g ses
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local abbreviation ex := SES.ex ses
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local abbreviation q := ab_qg_map (kernel_subgroup g)
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local abbreviation f' := SES.f ses'
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local abbreviation g' := SES.g ses'
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local notation `g'` := SES.g ses'
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local abbreviation ex' := SES.ex ses'
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local abbreviation q' := ab_qg_map (kernel_subgroup g')
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local abbreviation α := quotient_codomain_SES ses
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local abbreviation α' := quotient_codomain_SES ses'
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local abbreviation α' := quotient_codomain_SES
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include htpy1
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definition quotient_extend_unique_SES : is_contr (Σ (hC : C →g C'), hC ∘g g ~ g' ∘g hB) :=
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begin
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fapply @(is_trunc_equiv_closed_rev _ (quotient_triangle_extend_SES (g' ∘g hB))),
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fapply ab_qg_universal_property,
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intro b, intro K,
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have k : trunctype.carrier (image_subgroup f b), from is_exact.ker_in_im ex b K,
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induction k, induction a with a p,
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induction p,
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refine (ap g' (htpy1 a)) ⬝ _,
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fapply is_exact.im_in_ker ex' (hA a)
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end
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/-
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-- We define a group homomorphism B/ker(g) →g B'/ker(g'), keeping in mind that ker(g)=A and ker(g')=A'.
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definition quotient_extend_SES : quotient_ab_group (kernel_subgroup g) →g quotient_ab_group (kernel_subgroup g') :=
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begin
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@ -118,13 +170,8 @@ definition right_extend_SES : C →g C' :=
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local abbreviation hC := right_extend_SES
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definition right_extend_hom_SES : hom_SES ses ses' :=
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definition right_extend_SES_square : hC ∘g g ~ g' ∘ hB :=
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begin
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fapply hom_SES.mk,
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exact hA,
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exact hB,
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exact hC,
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exact htpy1,
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exact calc
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hC ∘g g ~ hC ∘g α ∘g q : by reflexivity
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... ~ α' ∘g k ∘g α⁻¹ᵍ ∘g α ∘g q : by reflexivity
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@ -133,4 +180,36 @@ definition right_extend_hom_SES : hom_SES ses ses' :=
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... ~ g' ∘g hB : by reflexivity
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end
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end short_exact_sequences
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local abbreviation htpy2 := right_extend_SES_square
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definition right_extend_SES_unique_map_aux (hC' : C →g C') (htpy2' : g' ∘g hB ~ hC' ∘g g) : k ∘g q ~ α'⁻¹ᵍ ∘g hC' ∘g α ∘g q :=
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begin
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exact calc
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k ∘g q ~ q' ∘g hB : by reflexivity
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... ~ α'⁻¹ᵍ ∘g α' ∘g q' ∘g hB : by exact hwhisker_right (q' ∘g hB) (homotopy.symm (left_inv α'))
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... ~ α'⁻¹ᵍ ∘g g' ∘g hB : by reflexivity
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... ~ α'⁻¹ᵍ ∘g hC' ∘g g : by exact hwhisker_left (α'⁻¹ᵍ) htpy2'
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... ~ α'⁻¹ᵍ ∘g hC' ∘g α ∘g q : by reflexivity
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end
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definition right_extend_SES_unique_map (hC' : C →g C') (htpy2' : hC' ∘g g ~ g' ∘g hB) : hC ~ hC' :=
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begin
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exact calc
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hC ~ α' ∘g k ∘g α⁻¹ᵍ : by reflexivity
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... ~ α' ∘g α'⁻¹ᵍ ∘g hC' ∘g α ∘g α⁻¹ᵍ :
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... ~ hC' ∘g α ∘g α⁻¹ᵍ : _
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... ~ hC' : _
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end
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definition right_extend_hom_SES : hom_SES ses ses' :=
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begin
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fapply hom_SES.mk,
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exact hA,
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exact hB,
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exact hC,
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exact htpy1,
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exact htpy2
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end
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-/
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end ses
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