AbelianGroup.v

Require Import Basics Types.

Require Export Classes.interfaces.canonical_names (Zero, zero, Plus).

Require Export Classes.interfaces.abstract_algebra (IsAbGroup(..), abgroup_group, abgroup_commutative).

Require Export Algebra.Groups.Group.

Require Export Algebra.Groups.Subgroup.

Require Import Algebra.Groups.QuotientGroup.

Require Import WildCat.


Local Set Polymorphic Inductive Cumulativity.


Local Open Scope mc_scope.

Local Open Scope mc_add_scope.


(** * Abelian groups *)

(** Definition of an abelian group *)

Record AbGroup := {
  abgroup_group : Group;
  abgroup_commutative : Commutative (@group_sgop abgroup_group);
}.


Coercion abgroup_group : AbGroup >-> Group.

Global Existing Instance abgroup_commutative.


Global Instance isabgroup_abgroup {A : AbGroup} : IsAbGroup A.

Proof.

  split; exact _.

Defined.


Definition issig_abgroup : _ <~> AbGroup := ltac:(issig).


Global Instance zero_abgroup (A : AbGroup) : Zero A := group_unit.

Global Instance plus_abgroup (A : AbGroup) : Plus A := group_sgop.

Global Instance negate_abgroup (A : AbGroup) : Negate A := group_inverse.


(** ** Paths between abelian groups *)

Definition equiv_path_abgroup `{Univalence} {A B : AbGroup@{u}}
  : GroupIsomorphism A B <~> (A = B).

Proof.

  refine (equiv_ap_inv issig_abgroup _ _ oE _).

  refine (equiv_path_sigma_hprop _ _ oE _).

  exact equiv_path_group.

Defined.


Definition equiv_path_abgroup_group `{Univalence} {A B : AbGroup}
  : (A = B :> AbGroup) <~> (A = B :> Group)
  := equiv_path_group oE equiv_path_abgroup^-1.


(** ** Subgroups of abelian groups *)

(** Subgroups of abelian groups are abelian *)

Global Instance isabgroup_subgroup (G : AbGroup) (H : Subgroup G)
  : IsAbGroup H.

Proof.

  nrapply Build_IsAbGroup.

  1: exact _.

  intros x y.

  apply path_sigma_hprop.

  cbn.

 apply commutativity.

Defined.


(** Given any subgroup of an abelian group, we can coerce it to an abelian group. Note that Coq complains this coercion doesn't satisfy the uniform-inheritance condition, but in practice it works and doesn't cause any issue, so we ignore it. *)

Definition abgroup_subgroup (G : AbGroup) : Subgroup G -> AbGroup
  := fun H => Build_AbGroup H _.

#[warnings="-uniform-inheritance"]
Coercion abgroup_subgroup : Subgroup >-> AbGroup.


Global Instance isnormal_ab_subgroup (G : AbGroup) (H : Subgroup G)
  : IsNormalSubgroup H.

Proof.

  intros x y; unfold in_cosetL, in_cosetR.

  refine (_ oE equiv_subgroup_inverse _ _).

  rewrite negate_sg_op.

  rewrite negate_involutive.

  by rewrite (commutativity (-y) x).

Defined.


(** ** Quotients of abelian groups *)

Global Instance isabgroup_quotient (G : AbGroup) (H : Subgroup G)
  : IsAbGroup (QuotientGroup' G H (isnormal_ab_subgroup G H)).

Proof.

  nrapply Build_IsAbGroup.

  1: exact _.

  intro x.

  srapply Quotient_ind_hprop.

  intro y; revert x.

  srapply Quotient_ind_hprop.

  intro x.

  apply (ap (class_of _)).

  apply commutativity.

Defined.


Definition QuotientAbGroup (G : AbGroup) (H : Subgroup G) : AbGroup
  := (Build_AbGroup (QuotientGroup' G H (isnormal_ab_subgroup G H)) _).


Definition quotient_abgroup_rec {G : AbGroup}
  (N : Subgroup G) (H : AbGroup)
  (f : GroupHomomorphism G H) (h : forall n : G, N n -> f n = mon_unit)
  : GroupHomomorphism (QuotientAbGroup G N) H
  := grp_quotient_rec G (Build_NormalSubgroup G N _) f h.


Theorem equiv_quotient_abgroup_ump {F : Funext} {G : AbGroup}
  (N : Subgroup G) (H : Group)
  : {f : GroupHomomorphism G H & forall (n : G), N n -> f n = mon_unit}
    <~> (GroupHomomorphism (QuotientAbGroup G N) H).

Proof.

  exact (equiv_grp_quotient_ump (Build_NormalSubgroup G N _) _).

Defined.


(** ** The wild category of abelian groups *)

Global Instance isgraph_abgroup : IsGraph AbGroup
  := isgraph_induced abgroup_group.


Global Instance is01cat_abgroup : Is01Cat AbGroup
  := is01cat_induced abgroup_group.


Global Instance is01cat_grouphomomorphism {A B : AbGroup} : Is01Cat (A $-> B)
  := is01cat_induced (@grp_homo_map A B).


Global Instance is0gpd_grouphomomorphism {A B : AbGroup} : Is0Gpd (A $-> B)
  := is0gpd_induced (@grp_homo_map A B).


Global Instance is2graph_abgroup : Is2Graph AbGroup
  := is2graph_induced abgroup_group.


(** AbGroup forms a 1Cat *)

Global Instance is1cat_abgroup : Is1Cat AbGroup
  := is1cat_induced _.


Global Instance hasmorext_abgroup `{Funext} : HasMorExt AbGroup
  := hasmorext_induced _.


Global Instance hasequivs_abgroup : HasEquivs AbGroup
  := hasequivs_induced _.


(** Zero object of AbGroup *)

Definition abgroup_trivial : AbGroup.

Proof.

  rapply (Build_AbGroup grp_trivial).

  by intros [].

Defined.


(** AbGroup is a pointed category *)

Global Instance ispointedcat_abgroup : IsPointedCat AbGroup.

Proof.

  apply Build_IsPointedCat with abgroup_trivial.

  all: intro A; apply ispointedcat_group.

Defined.


(** Image of group homomorphisms between abelian groups *)

Definition abgroup_image {A B : AbGroup} (f : A $-> B) : AbGroup
  := Build_AbGroup (grp_image f) _.


(** First isomorphism theorem of abelian groups *)

Definition abgroup_first_iso `{Funext} {A B : AbGroup} (f : A $-> B)
  : GroupIsomorphism (QuotientAbGroup A (grp_kernel f)) (abgroup_image f).

Proof.

  etransitivity.

  2: rapply grp_first_iso.

  apply grp_iso_quotient_normal.

Defined.


(** ** Kernels of abelian groups *)

Definition ab_kernel {A B : AbGroup} (f : A $-> B) : AbGroup
  := Build_AbGroup (grp_kernel f) _.


(** ** Transporting in families related to abelian groups *)

Lemma transport_abgrouphomomorphism_from_const `{Univalence} {A B B' : AbGroup}
      (p : B = B') (f : GroupHomomorphism A B)
  : transport (Hom A) p f
    = grp_homo_compose (equiv_path_abgroup^-1 p) f.

Proof.

  induction p.

  by apply equiv_path_grouphomomorphism.

Defined.


Lemma transport_iso_abgrouphomomorphism_from_const `{Univalence} {A B B' : AbGroup}
      (phi : GroupIsomorphism B B') (f : GroupHomomorphism A B)
  : transport (Hom A) (equiv_path_abgroup phi) f
    = grp_homo_compose phi f.

Proof.

  refine (transport_abgrouphomomorphism_from_const _ _ @ _).

  by rewrite eissect.

Defined.


Lemma transport_abgrouphomomorphism_to_const `{Univalence} {A A' B : AbGroup}
      (p : A = A') (f : GroupHomomorphism A B)
  : transport (fun G => Hom G B) p f
    = grp_homo_compose f (grp_iso_inverse (equiv_path_abgroup^-1 p)).

Proof.

  induction p; cbn.

  by apply equiv_path_grouphomomorphism.

Defined.


Lemma transport_iso_abgrouphomomorphism_to_const `{Univalence} {A A' B : AbGroup}
      (phi : GroupIsomorphism A A') (f : GroupHomomorphism A B)
  : transport (fun G => Hom G B) (equiv_path_abgroup phi) f
    = grp_homo_compose f (grp_iso_inverse phi).

Proof.

  refine (transport_abgrouphomomorphism_to_const _ _ @ _).

  by rewrite eissect.

Defined.


(** ** Operations on abelian groups *)

(** The negation automorphism of an abelian group *)

Definition ab_homo_negation {A : AbGroup} : GroupIsomorphism A A.

Proof.

  snrapply Build_GroupIsomorphism.

 snrapply Build_GroupHomomorphism.

 exact (fun a => -a).

 intros x y.

      refine (grp_inv_op x y @ _).

      apply commutativity.

 srapply isequiv_adjointify.

    1: exact (fun a => -a).

    1-2: exact negate_involutive.

Defined.


(** Multiplication by [n : nat] defines an endomorphism of any abelian group [A]. *)

Definition ab_mul_nat {A : AbGroup} (n : nat) : GroupHomomorphism A A.

Proof.

  snrapply Build_GroupHomomorphism.

  1: exact (fun a => grp_pow a n).

  intros a b.

  induction n; cbn.

  1: exact (grp_unit_l _)^.

  refine (_ @ associativity _ _ _).

  refine (_ @ ap _ (associativity _ _ _)^).

  rewrite (commutativity (grp_pow a n) b).

  refine (_ @ ap _ (associativity _ _ _)).

  refine (_ @ (associativity _ _ _)^).

  apply grp_cancelL.

  assumption.

Defined.


Definition ab_mul_nat_homo {A B : AbGroup}
  (f : GroupHomomorphism A B) (n : nat)
  : f o ab_mul_nat n == ab_mul_nat n o f
  := grp_pow_homo f n.


(** The image of an inclusion is a normal subgroup. *)

Definition ab_image_embedding {A B : AbGroup} (f : A $-> B) `{IsEmbedding f} : NormalSubgroup B
  := {| normalsubgroup_subgroup := grp_image_embedding f; normalsubgroup_isnormal := _ |}.


Definition ab_image_in_embedding {A B : AbGroup} (f : A $-> B) `{IsEmbedding f}
  : GroupIsomorphism A (ab_image_embedding f)
  := grp_image_in_embedding f.


(** The cokernel of a homomorphism into an abelian group. *)

Definition ab_cokernel {G : Group@{u}} {A : AbGroup@{u}} (f : GroupHomomorphism G A)
  : AbGroup := QuotientAbGroup _ (grp_image f).


Definition ab_cokernel_embedding {G : Group} {A : AbGroup} (f : G $-> A) `{IsEmbedding f}
  : AbGroup := QuotientAbGroup _ (grp_image_embedding f).


Definition ab_cokernel_embedding_rec {G: Group} {A B : AbGroup} (f : G $-> A) `{IsEmbedding f}
  (h : A $-> B) (p : grp_homo_compose h f $== grp_homo_const)
  : ab_cokernel_embedding f $-> B.

Proof.

  snrapply (grp_quotient_rec _ _ h).

  intros a [g q].

  induction q.

  exact (p g).

Defined.

Timings for AbelianGroup.v