Timings for Biproduct.v

Require Import Basics Types Truncations.Core.

Require Import WildCat.

Require Import HSet.

Require Import AbelianGroup.

Require Import Modalities.ReflectiveSubuniverse.

Local Open Scope mc_add_scope.

(** * Biproducts of abelian groups *)

Definition ab_biprod@{u} (A B : AbGroup@{u}) : AbGroup@{u}.

Proof.

  rapply (Build_AbGroup (grp_prod A B)).

  intros [a b] [a' b'].

  apply path_prod; simpl; apply commutativity.

Defined.

(** These inherit [IsEmbedding] instances from their [grp_prod] versions. *)

Definition ab_biprod_inl {A B : AbGroup} : A $-> ab_biprod A B := grp_prod_inl.

Definition ab_biprod_inr {A B : AbGroup} : B $-> ab_biprod A B := grp_prod_inr.

(** These inherit [IsSurjection] instances from their [grp_prod] versions. *)

Definition ab_biprod_pr1 {A B : AbGroup} : ab_biprod A B $-> A := grp_prod_pr1.

Definition ab_biprod_pr2 {A B : AbGroup} : ab_biprod A B $-> B := grp_prod_pr2.

(** Recursion principle *)

Proposition ab_biprod_rec {A B Y : AbGroup}
            (f : A $-> Y) (g : B $-> Y)
  : (ab_biprod A B) $-> Y.

Proof.

  snrapply Build_GroupHomomorphism.

  -

 intros [a b]; exact (f a + g b).

  -

 intros [a b] [a' b']; simpl.

    rewrite (grp_homo_op f).

    rewrite (grp_homo_op g).

    rewrite (associativity _ (g b) _).

    rewrite <- (associativity _ (f a') _).

    rewrite (commutativity (f a') _).

    rewrite (associativity _ (g b) _).

    exact (associativity _ (f a') _)^.

Defined.

Corollary ab_biprod_rec_uncurried {A B Y : AbGroup}
  : (A $-> Y) * (B $-> Y)
    -> (ab_biprod A B) $-> Y.

Proof.

  intros [f g].

 exact (ab_biprod_rec f g).

Defined.

Proposition ab_biprod_rec_eta {A B Y : AbGroup}
            (u : ab_biprod A B $-> Y)
  : ab_biprod_rec (u $o ab_biprod_inl) (u $o ab_biprod_inr) == u.

Proof.

  intros [a b]; simpl.

  refine ((grp_homo_op u _ _)^ @ ap u _).

  apply path_prod.

  -

 exact (right_identity a).

  -

 exact (left_identity b).

Defined.

Proposition ab_biprod_rec_inl_beta {A B Y : AbGroup}
            (a : A $-> Y) (b : B $-> Y)
  : (ab_biprod_rec a b) $o ab_biprod_inl == a.

Proof.

  intro x; simpl.

  rewrite (grp_homo_unit b).

  exact (right_identity (a x)).

Defined.

Proposition ab_biprod_rec_inr_beta {A B Y : AbGroup}
            (a : A $-> Y) (b : B $-> Y)
  : (ab_biprod_rec a b) $o ab_biprod_inr == b.

Proof.

  intro y; simpl.

  rewrite (grp_homo_unit a).

  exact (left_identity (b y)).

Defined.

Theorem isequiv_ab_biprod_rec `{Funext} {A B Y : AbGroup}
  : IsEquiv (@ab_biprod_rec_uncurried A B Y).

Proof.

  srapply isequiv_adjointify.

  -

 intro phi.

    exact (phi $o ab_biprod_inl, phi $o ab_biprod_inr).

  -

 intro phi.

    apply equiv_path_grouphomomorphism.

    exact (ab_biprod_rec_eta phi).

  -

 intros [a b].

    apply path_prod.

    +

 apply equiv_path_grouphomomorphism.

      apply ab_biprod_rec_inl_beta.

    +

 apply equiv_path_grouphomomorphism.

      apply ab_biprod_rec_inr_beta.

Defined.

(** Corecursion principle, inherited from Groups/Group.v. *)

Definition ab_biprod_corec {A B X : AbGroup} (f : X $-> A) (g : X $-> B)
  : X $-> ab_biprod A B
  := grp_prod_corec f g.

Definition ab_corec_eta {X Y A B : AbGroup} (f : X $-> Y) (g0 : Y $-> A) (g1 : Y $-> B)
  : ab_biprod_corec g0 g1 $o f $== ab_biprod_corec (g0 $o f) (g1 $o f)
  := fun _ => idpath.

(** *** Functoriality of [ab_biprod] *)

Definition functor_ab_biprod {A A' B B' : AbGroup} (f : A $-> A') (g: B $-> B')
  : ab_biprod A B $-> ab_biprod A' B'
  := (ab_biprod_corec (f $o ab_biprod_pr1) (g $o ab_biprod_pr2)).

Definition ab_biprod_functor_beta {Z X Y A B : AbGroup} (f0 : Z $-> X) (f1 : Z $-> Y)
           (g0 : X $-> A) (g1 : Y $-> B)
  : functor_ab_biprod g0 g1 $o ab_biprod_corec f0 f1
                      $== ab_biprod_corec (g0 $o f0) (g1 $o f1)
  := fun _ => idpath.

Definition isequiv_functor_ab_biprod {A A' B B' : AbGroup}
           (f : A $-> A') (g : B $-> B') `{IsEquiv _ _ f} `{IsEquiv _ _ g}
  : IsEquiv (functor_ab_biprod f g).

Proof.

  srapply isequiv_adjointify.

  1: {

 rapply functor_ab_biprod;
       apply grp_iso_inverse.

       +

 exact (Build_GroupIsomorphism _ _ f _).

       +

 exact (Build_GroupIsomorphism _ _ g _).

 }

  all: intros [a b]; simpl.

  all: apply path_prod'.

  1,2: apply eisretr.

  all: apply eissect.

Defined.

Definition equiv_functor_ab_biprod {A A' B B' : AbGroup}
           (f : A $-> A') (g : B $-> B') `{IsEquiv _ _ f} `{IsEquiv _ _ g}
  : GroupIsomorphism (ab_biprod A B) (ab_biprod A' B')
  := Build_GroupIsomorphism _ _ _ (isequiv_functor_ab_biprod f g).

(** Biproducts preserve embeddings. *)

Definition functor_ab_biprod_embedding {A A' B B' : AbGroup}
           (i : A $-> B) (i' : A' $-> B')
           `{IsEmbedding i} `{IsEmbedding i'}
  : IsEmbedding (functor_ab_biprod i i').

Proof.

  intros [b b'].

  apply hprop_allpath.

  intros [[a0 a0'] p] [[a1 a1'] p']; cbn in p, p'.

  rapply path_sigma_hprop; cbn.

  pose (q := (equiv_path_prod _ _)^-1 p); cbn in q.

  pose (q' := (equiv_path_prod _ _)^-1 p'); cbn in q'.

  destruct q as [q0 q1], q' as [q0' q1'].

  apply path_prod; rapply isinj_embedding; cbn.

  -

 exact (q0 @ q0'^).

  -

 exact (q1 @ q1'^).

Defined.

(** Products preserve surjections. *)

Definition functor_ab_biprod_surjection `{Funext} {A A' B B' : AbGroup}
           (p : A $-> B) (p' : A' $-> B')
           `{S : IsSurjection p} `{S' : IsSurjection p'}
  : IsSurjection (functor_ab_biprod p p').

Proof.

  intros [b b'].

  pose proof (a := S b); pose proof (a' := S' b').

  apply center in a, a'.

  strip_truncations.

  rapply contr_inhabited_hprop.

  apply tr.

  exists (ab_biprod_inl a.1 + ab_biprod_inr a'.1); cbn.

  apply path_prod;
    refine (grp_homo_op _ _ _ @ _);
    rewrite (grp_homo_unit _);
    cbn.

  -

 exact (right_identity _ @ a.2).

  -

 exact (left_identity _ @ a'.2).

Defined.

(** *** Lemmas for working with biproducts *)

Lemma ab_biprod_decompose {B A : AbGroup} (a : A) (b : B)
  : (a, b) = ((a, group_unit) : ab_biprod A B) + (group_unit, b).

Proof.

  apply path_prod; cbn.

  -

 exact (right_identity _)^.

  -

 exact (left_identity _)^.

Defined.

Lemma ab_biprod_corec_eta' {A B X : AbGroup} (f g : ab_biprod A B $-> X)
  : (f $o ab_biprod_inl $== g $o ab_biprod_inl)
  -> (f $o ab_biprod_inr $== g $o ab_biprod_inr)
  -> f $== g.

Proof.

  intros h k.

  intros [a b].

  refine (ap f (ab_biprod_decompose _ _) @ _ @ ap g (ab_biprod_decompose _ _)^).

  refine (grp_homo_op _ _ _ @ _ @ (grp_homo_op _ _ _)^).

  exact (ap011 (+) (h a) (k b)).

Defined.

(* Maps out of biproducts are determined on the two inclusions. *)

Lemma equiv_path_biprod_corec `{Funext} {A B X : AbGroup} (phi psi : ab_biprod A B $-> X)
  : ((phi $o ab_biprod_inl == psi $o ab_biprod_inl) * (phi $o ab_biprod_inr == psi $o ab_biprod_inr))
      <~> phi == psi.

Proof.

  apply equiv_iff_hprop.

  -

 intros [h k].

    apply ab_biprod_corec_eta'; assumption.

  -

 intro h.

    exact (fun a => h _, fun b => h _).

Defined.

(** The swap isomorphism of the biproduct of two groups. *)

Definition direct_sum_swap {A B : AbGroup}
  : ab_biprod A B $<~> ab_biprod B A.

Proof.

  snrapply Build_GroupIsomorphism'.

  -

 apply equiv_prod_symm.

  -

 intro; reflexivity.

Defined.

(** Addition [+] is a group homomorphism [A+A -> A]. *)

Definition ab_codiagonal {A : AbGroup} : ab_biprod A A $-> A
  := ab_biprod_rec grp_homo_id grp_homo_id.

Definition ab_codiagonal_natural {A B : AbGroup} (f : A $-> B)
  : f $o ab_codiagonal $== ab_codiagonal $o functor_ab_biprod f f
  := fun a => grp_homo_op f _ _.

Definition ab_diagonal {A : AbGroup} : A $-> ab_biprod A A
  := ab_biprod_corec grp_homo_id grp_homo_id.

(** Given two abelian group homomorphisms [f] and [g], their pairing [(f, g) : B -> A + A] can be written as a composite. Note that [ab_biprod_corec] is an alias for [grp_prod_corec]. *)

Lemma ab_biprod_corec_diagonal `{Funext} {A B : AbGroup} (f g : B $-> A)
  : ab_biprod_corec f g = (functor_ab_biprod f g) $o ab_diagonal.

Proof.

  apply equiv_path_grouphomomorphism; reflexivity.

Defined.

(** Precomposing the codiagonal with the swap map has no effect. *)

Lemma ab_codiagonal_swap `{Funext} {A : AbGroup}
  : (@ab_codiagonal A) $o direct_sum_swap = ab_codiagonal.

Proof.

  apply equiv_path_grouphomomorphism.

  intro a; cbn.

  exact (abgroup_commutative _ _ _).

Defined.

(** The corresponding result for the diagonal is true definitionally, so it isn't strictly necessary to state it, but we record it anyways. *)

Definition ab_diagonal_swap {A : AbGroup}
  : direct_sum_swap $o (@ab_diagonal A) = ab_diagonal
  := idpath.

(** The biproduct is associative. *)

Lemma ab_biprod_assoc {A B C : AbGroup}
  : ab_biprod A (ab_biprod B C) $<~> ab_biprod (ab_biprod A B) C.

Proof.

  snrapply Build_GroupIsomorphism'.

  -

 apply equiv_prod_assoc.

  -

 unfold IsSemiGroupPreserving; reflexivity.

Defined.

(** The iterated diagonals [(ab_diagonal + id) o ab_diagonal] and [(id + ab_diagonal) o ab_diagonal] agree, after reassociating the direct sum. *)

Definition ab_commute_id_diagonal {A : AbGroup}
  : (functor_ab_biprod (@ab_diagonal A) grp_homo_id) $o ab_diagonal
    = ab_biprod_assoc $o (functor_ab_biprod grp_homo_id ab_diagonal) $o ab_diagonal
  := idpath.

(** A similar result for the codiagonal. *)

Lemma ab_commute_id_codiagonal `{Funext} {A : AbGroup}
  : (@ab_codiagonal A) $o (functor_ab_biprod ab_codiagonal grp_homo_id) $o ab_biprod_assoc
    = ab_codiagonal $o (functor_ab_biprod grp_homo_id ab_codiagonal).

Proof.

  apply equiv_path_grouphomomorphism.

  intro a; cbn.

  exact (grp_assoc _ _ _)^.

Defined.

(** The next few results are used to prove associativity of the Baer sum. *)

(** A "twist" isomorphism [(A + B) + C <~> (C + B) + A]. *)

Lemma ab_biprod_twist {A B C : AbGroup@{u}}
  : ab_biprod (ab_biprod A B) C $<~> ab_biprod (ab_biprod C B) A.

Proof.

  snrapply Build_GroupIsomorphism.

  -

 snrapply Build_GroupHomomorphism.

    +

 intros [[a b] c].

      exact ((c,b),a).

    +

 unfold IsSemiGroupPreserving.

 reflexivity.

  -

 snrapply isequiv_adjointify.

    +

 intros [[c b] a].

      exact ((a,b),c).

    +

 reflexivity.

    +

 reflexivity.

Defined.

(** The triagonal and cotriagonal homomorphisms. *)

Definition ab_triagonal {A : AbGroup} : A $-> ab_biprod (ab_biprod A A) A
  := (functor_ab_biprod ab_diagonal grp_homo_id) $o ab_diagonal.

Definition ab_cotriagonal {A : AbGroup} : ab_biprod (ab_biprod A A) A $-> A
  := ab_codiagonal $o (functor_ab_biprod ab_codiagonal grp_homo_id).

(** For an abelian group [A], precomposing the triagonal on [A] with the twist map on [A] has no effect. *)

Definition ab_triagonal_twist {A : AbGroup}
  : ab_biprod_twist $o @ab_triagonal A = ab_triagonal
  := idpath.

(** A similar result for the cotriagonal. *)

Definition ab_cotriagonal_twist `{Funext} {A : AbGroup}
  : @ab_cotriagonal A $o ab_biprod_twist = ab_cotriagonal.

Proof.

  apply equiv_path_grouphomomorphism.

  intro x.

 cbn.

  refine ((grp_assoc _ _ _)^ @ _).

  refine (abgroup_commutative _ _ _ @ _).

  exact (ap (fun a =>  a + snd x) (abgroup_commutative _ _ _)).

Defined.