TensorProduct.v


(** * The Tensor Product of Abelian Groups *)

(** Various maps [A * B → C] from the cartesian product of two abelian groups to another abelian group are "biadditive" (also called "bilinear"), meaning that they are group homomorphisms when we fix the left or right argument.

The tensor product of abelian groups is a construction that produces an abelian group [A ⊗ B] along with a biadditive map [A * B -> A ⊗ B] which is initial among biadditive maps from [A * B].  This means that any biadditive map [A * B → C] factors uniquely through the tensor product via a group homomorphism [A ⊗ B -> C].

Biadditive functions appear in all sorts of contexts ranging from linear algebra to analysis. Therefore having a way to systematically study them is very useful. *)

(** ** Construction *)

(** We define the tensor product of abelian groups as a quotient of the free abelian group on pairs of elements of the two groups by the subgroup generated by the biadditive pairs. *)

(** Here we define the subgroup of biadditive pairs in two steps. *)

Definition family_biadditive_pairs {A B : AbGroup}
  : FreeAbGroup (A * B) -> Type.

  refine ((exists (a1 a2 : A) (b : B), _) + exists (a : A) (b1 b2 : B), _)%type.

 refine (- _ + (_ + _) = x).

    1-3: apply freeabgroup_in.

 refine (- _ + (_ + _) = x).

    1-3: apply freeabgroup_in.


Definition subgroup_biadditive_pairs {A B : AbGroup}
  : Subgroup (FreeAbGroup (A * B))
  := subgroup_generated family_biadditive_pairs.


(** The tensor product [ab_tensor_prod A B] of two abelian groups [A] and [B] is defined to be a quotient of the free abelian group on pairs of elements [A * B] by the subgroup of biadditive pairs. *)

Definition ab_tensor_prod (A B : AbGroup) : AbGroup
  := QuotientAbGroup (FreeAbGroup (A * B)) subgroup_biadditive_pairs.


Arguments ab_tensor_prod A B : simpl never.


(** The tensor product of [A] and [B] contains formal sums and differences of pairs of elements from [A] and [B]. We denote these pairs as "simple tensors" and name them [tensor]. *)

Definition tensor {A B : AbGroup} : A -> B -> ab_tensor_prod A B
  := fun a b => grp_quotient_map (freeabgroup_in (a, b)).


(** ** Properties of tensors *)

(** The characterizing property of simple tensors are that they are biadditive in their arguments. *)

(** A [tensor] of a sum distributes over the sum on the left. *)

Definition tensor_dist_l {A B : AbGroup} (a : A) (b b' : B)
  : tensor a (b + b') = tensor a b + tensor a b'.


(** A [tensor] of a sum distributes over the sum on the right. *)

Definition tensor_dist_r {A B : AbGroup} (a a' : A) (b : B)
  : tensor (a + a') b = tensor a b + tensor a' b.


(** Tensoring on the left is a group homomorphism. *)

Definition grp_homo_tensor_l {A B : AbGroup} (a : A)
  : B $-> ab_tensor_prod A B.

  snapply Build_GroupHomomorphism.

 exact (fun b => tensor a b).

    napply tensor_dist_l.

 

(** Tensoring on the right is a group homomorphism. *)

Definition grp_homo_tensor_r {A B : AbGroup} (b : B)
  : A $-> ab_tensor_prod A B.

  snapply Build_GroupHomomorphism.

 exact (fun a => tensor a b).

    napply tensor_dist_r.


(** Tensors preserve negation in the left argument. *)

Definition tensor_neg_l {A B : AbGroup} (a : A) (b : B)
  : tensor (-a) b = - tensor a b
  := grp_homo_inv (grp_homo_tensor_r b) a.


(** Tensors preserve negation in the right argument. *)

Definition tensor_neg_r {A B : AbGroup} (a : A) (b : B)
  : tensor a (-b) = - tensor a b
  := grp_homo_inv (grp_homo_tensor_l a) b.


(** Tensoring by zero on the left is zero. *)

Definition tensor_zero_l {A B : AbGroup} (b : B)
  : tensor (A:=A) 0 b = 0
  := grp_homo_unit (grp_homo_tensor_r b).


(** Tensoring by zero on the right is zero. *)

Definition tensor_zero_r {A B : AbGroup} (a : A)
  : tensor (B:=B) a 0 = 0
  := grp_homo_unit (grp_homo_tensor_l a).


(** The [tensor] map is biadditive and therefore can be written in a curried form using the internal abelian group hom. *)

Definition grp_homo_tensor `{Funext} {A B : AbGroup}
  : A $-> ab_hom B (ab_tensor_prod A B).

  snapply Build_GroupHomomorphism.

    snapply Build_GroupHomomorphism.

 napply tensor_dist_l.

    apply equiv_path_grouphomomorphism.

    napply tensor_dist_r.


(** ** Induction principles *)

(** Here we write down some induction principles to help us prove lemmas about the tensor product. Some of these are quite specialised but are patterns that appear often in practice. *) 

(** Our main recursion principle states that in order to build a homomorphism out of the tensor product, it is sufficient to provide a map out of the direct product which is biadditive, that is, a map that preserves addition in each argument of the product. *)

(** We separate out the proof of this part, so we can make it opaque. *)

Definition ab_tensor_prod_rec_helper {A B C : AbGroup}
  (f : A -> B -> C)
  (l : forall a b b', f a (b + b') = f a b + f a b')
  (r : forall a a' b, f (a + a') b = f a b + f a' b)
  (x : FreeAbGroup (A * B)) (insg : subgroup_biadditive_pairs x)
  : grp_homo_abel_rec (FreeGroup_rec (uncurry f)) x = mon_unit.

  set (abel_rec := grp_homo_abel_rec (FreeGroup_rec (uncurry f))).

  induction insg as [ x biad | | g h insg_g IHg insg_h IHh ].

 destruct biad as [ [ a [ a' [ b p ] ] ] | [ a [ b [ b' p ] ] ] ].

    all: destruct p; simpl.

    all: apply grp_moveL_M1^-1%equiv; symmetry.

 napply grp_homo_unit.

 rewrite grp_homo_op, grp_homo_inv.

    apply grp_moveL_1M^-1.


Opaque ab_tensor_prod_rec_helper.


Definition ab_tensor_prod_rec {A B C : AbGroup}
  (f : A -> B -> C)
  (l : forall a b b', f a (b + b') = f a b + f a b')
  (r : forall a a' b, f (a + a') b = f a b + f a' b) 
  : ab_tensor_prod A B $-> C.

  unfold ab_tensor_prod.

  snapply grp_quotient_rec.

 snapply FreeAbGroup_rec.

 unfold normalsubgroup_subgroup.

    apply ab_tensor_prod_rec_helper; assumption.


(** A special case that arises. *)

Definition ab_tensor_prod_rec' {A B C : AbGroup}
  (f : A -> (B $-> C))
  (l : forall a a' b, f (a + a') b = f a b + f a' b)
  : ab_tensor_prod A B $-> C.

  refine (ab_tensor_prod_rec f _ l).

  intro a; apply grp_homo_op.


(** We give an induction principle for an hprop-valued type family [P].  It may be surprising at first that we only require [P] to hold for the simple tensors [tensor a b] and be closed under addition.  It automatically follows that [P 0] holds (since [tensor 0 0 = 0]) and that [P] is closed under negation (since [tensor -a b = - tensor a b]). This induction principle says that the simple tensors generate the tensor product as a semigroup. *)

Definition ab_tensor_prod_ind_hprop {A B : AbGroup}
  (P : ab_tensor_prod A B -> Type)
  {H : forall x, IsHProp (P x)}
  (Hin : forall a b, P (tensor a b))
  (Hop : forall x y, P x -> P y -> P (x + y))
  : forall x, P x.

  unfold ab_tensor_prod.

  srapply grp_quotient_ind_hprop.

  srapply Abel_ind_hprop; cbn beta.

  set (tensor_in := grp_quotient_map $o abel_unit : FreeGroup (A * B) $-> ab_tensor_prod A B).

  change (forall x, P (tensor_in x)).

  srapply FreeGroup_ind_hprop'; intros w; cbn beta.

 (* The goal here is [P 0], so we use [Hin 0 0 : P (tensor 0 0)]. *)

exact (transport P (tensor_zero_l 0) (Hin 0 0)).

 change (P (tensor_in (freegroup_eta [a]%list + freegroup_eta w))).

    (* This [rewrite] is [reflexivity], but the [Defined] is slow if [change] is used instead. *)

rewrite grp_homo_op.

    destruct a as [[a b]|[a b]].

 change (P (tensor_in (freegroup_in (a, b)) + tensor_in (freegroup_eta w))).

 change (P (tensor_in (- freegroup_in (a, b)) + tensor_in (freegroup_eta w))).

      (* This [rewrite] is reflexivity, but using [change] to achieve this is slow and slows down the Defined line. *)

rewrite grp_homo_inv.

      rewrite <- tensor_neg_l.


(** As a commonly occurring special case of the above induction principle, we have the case when the predicate in question is showing that two group homomorphisms out of the tensor product are homotopic. In order to do this, it suffices to show it only for simple tensors. The homotopy is closed under addition, so we don't need to hypothesise anything else. *)

Definition ab_tensor_prod_ind_homotopy {A B G : AbGroup}
  {f f' : ab_tensor_prod A B $-> G}
  (H : forall a b, f (tensor a b) = f' (tensor a b))
  : f $== f'.

  napply ab_tensor_prod_ind_hprop.

 intros x y; apply grp_homo_op_agree.


(** As an even more specialised case, we occasionally have the second homomorphism being a sum of abelian group homomorphisms. In those cases, it is easier to use this specialised lemma. *)

Definition ab_tensor_prod_ind_homotopy_plus {A B G : AbGroup}
  {f f' f'' : ab_tensor_prod A B $-> G}
  (H : forall a b, f (tensor a b) = f' (tensor a b) + f'' (tensor a b))
  : forall x, f x = f' x + f'' x
  := ab_tensor_prod_ind_homotopy (f':=f' + f'') H.


(** Here we give an induction principle for a triple tensor, a.k.a a dependent trilinear function. *)

Definition ab_tensor_prod_ind_hprop_triple {A B C : AbGroup}
  (P : ab_tensor_prod A (ab_tensor_prod B C) -> Type)
  (H : forall x, IsHProp (P x))
  (Hin : forall a b c, P (tensor a (tensor b c)))
  (Hop : forall x y, P x -> P y -> P (x + y))
  : forall x, P x.

  rapply (ab_tensor_prod_ind_hprop P).

    rapply (ab_tensor_prod_ind_hprop (fun x => P (tensor _ x))).

      rewrite tensor_dist_l.


(** Similar to before, we specialise the triple tensor induction principle for proving homotopies of trilinear/triadditive functions. *)

Definition ab_tensor_prod_ind_homotopy_triple {A B C G : AbGroup}
  {f f' : ab_tensor_prod A (ab_tensor_prod B C) $-> G}
  (H : forall a b c, f (tensor a (tensor b c)) = f' (tensor a (tensor b c)))
  : f $== f'.

  napply ab_tensor_prod_ind_hprop_triple.

 intros x y; apply grp_homo_op_agree.


(** As explained for the biadditive and triadditive cases, we also derive an induction principle for quadruple tensors giving us dependent quadrilinear maps. *)

Definition ab_tensor_prod_ind_hprop_quad {A B C D : AbGroup}
  (P : ab_tensor_prod A (ab_tensor_prod B (ab_tensor_prod C D)) -> Type)
  (H : forall x, IsHProp (P x))
  (Hin : forall a b c d, P (tensor a (tensor b (tensor c d))))
  (Hop : forall x y, P x -> P y -> P (x + y))
  : forall x, P x.

  rapply (ab_tensor_prod_ind_hprop P).

    napply (ab_tensor_prod_ind_hprop_triple (fun x => P (tensor _ x))).

      rewrite tensor_dist_l.


(** To construct a homotopy between quadrilinear maps we need only check equality for the quadruple simple tensors. *)

Definition ab_tensor_prod_ind_homotopy_quad {A B C D G : AbGroup}
  {f f' : ab_tensor_prod A (ab_tensor_prod B (ab_tensor_prod C D)) $-> G}
  (H : forall a b c d, f (tensor a (tensor b (tensor c d)))
    = f' (tensor a (tensor b (tensor c d))))
  : f $== f'.

  napply (ab_tensor_prod_ind_hprop_quad (fun _ => _)).

 intros x y; apply grp_homo_op_agree.


(** ** Universal Property of the Tensor Product *)

(** A function of two variables is biadditive if it preserves the operation in each variable. *)

Class IsBiadditive {A B C : Type} `{SgOp A, SgOp B, SgOp C} (f : A -> B -> C) := {
  isbiadditive_l :: forall b, IsSemiGroupPreserving (flip f b);
  isbiadditive_r :: forall a, IsSemiGroupPreserving (f a);  
}.


Definition issig_IsBiadditive {A B C : Type} `{SgOp A, SgOp B, SgOp C}
  (f : A -> B -> C)
  : _ <~> IsBiadditive f
  := ltac:(issig).


(** The truncation level of the [IsBiadditive f] predicate is determined by the truncation level of the codomain. This will almost always be a hset. *)

Instance istrunc_isbiadditive `{Funext}
  {A B C : Type} `{SgOp A, SgOp B, SgOp C}
  (f : A -> B -> C) n `{IsTrunc n.+1 C}
  : IsTrunc n (IsBiadditive f).

  napply istrunc_equiv_istrunc.

  1: rapply issig_IsBiadditive.

  unfold IsSemiGroupPreserving.


(** The simple tensor map is biadditive. *)

Instance isbiadditive_tensor (A B : AbGroup)
  : IsBiadditive (@tensor A B) := {|
  isbiadditive_l := fun b a a' => tensor_dist_r a a' b;
  isbiadditive_r := tensor_dist_l;
|}.


(** The type of biadditive maps. *)

Record Biadditive (A B C : Type) `{SgOp A, SgOp B, SgOp C} := {
  biadditive_fun :> A -> B -> C;
  biadditive_isbiadditive :: IsBiadditive biadditive_fun;
}.


Definition issig_Biadditive {A B C : Type} `{SgOp A, SgOp B, SgOp C}
  : _ <~> Biadditive A B C
  := ltac:(issig).


Definition biadditive_ab_tensor_prod {A B C : AbGroup}
  : (ab_tensor_prod A B $-> C) -> Biadditive A B C.

  exists (fun x y => f (tensor x y)).

  snapply Build_IsBiadditive.

 intros b a a'; simpl.

    1: napply tensor_dist_r.

 intros a a' b; simpl.

    1: napply tensor_dist_l.


(** The universal property of the tensor product is that biadditive maps between abelian groups are in one-to-one correspondence with maps out of the tensor product. In this sense, the tensor product is the most perfect object describing biadditive maps between two abelian groups. *)

Definition equiv_ab_tensor_prod_rec `{Funext} (A B C : AbGroup)
  : Biadditive A B C <~> (ab_tensor_prod A B $-> C).

  snapply equiv_adjointify.

    exact (ab_tensor_prod_rec f r (fun a a' b => l b a a')).

 snapply biadditive_ab_tensor_prod.

    snapply equiv_path_grouphomomorphism.

    snapply ab_tensor_prod_ind_homotopy.

    snapply (equiv_ap_inv' issig_Biadditive).

    rapply path_sigma_hprop; simpl.


(** ** Functoriality of the Tensor Product *)

(** The tensor product produces a bifunctor and we will later show that it gives a symmetric monoidal structure on the category of abelian groups. *)

(** Given a pair of maps, we can produce a homomorphism between the pairwise tensor products of the domains and codomains. *)

Definition functor_ab_tensor_prod {A B A' B' : AbGroup}
  (f : A $-> A') (g : B $-> B')
  : ab_tensor_prod A B $-> ab_tensor_prod A' B'.

  snapply ab_tensor_prod_rec'.

    exact (grp_homo_tensor_l (f a) $o g).

    rewrite grp_homo_op.

    napply tensor_dist_r.


(** 2-functoriality of the tensor product. *)

Definition functor2_ab_tensor_prod {A B A' B' : AbGroup}
  {f f' : A $-> A'} (p : f $== f') {g g' : B $-> B'} (q : g $== g')
  : functor_ab_tensor_prod f g $== functor_ab_tensor_prod f' g'.

  snapply ab_tensor_prod_ind_homotopy.

  exact (ap011 tensor (p _) (q _)).


(** The tensor product functor preserves identity morphisms. *)

Definition functor_ab_tensor_prod_id (A B : AbGroup)
  : functor_ab_tensor_prod (Id A) (Id B) $== Id (ab_tensor_prod A B).

  snapply ab_tensor_prod_ind_homotopy.


(** The tensor product functor preserves composition. *)

Definition functor_ab_tensor_prod_compose {A B C A' B' C' : AbGroup}
  (f : A $-> B) (g : B $-> C) (f' : A' $-> B') (g' : B' $-> C')
  : functor_ab_tensor_prod (g $o f) (g' $o f')
    $== functor_ab_tensor_prod g g' $o functor_ab_tensor_prod f f'.

  snapply ab_tensor_prod_ind_homotopy.


(** The tensor product functor is a 0-bifunctor. *)

Instance is0bifunctor_ab_tensor_prod : Is0Bifunctor ab_tensor_prod.

  rapply Build_Is0Bifunctor'.

  snapply Build_Is0Functor.

  intros [A B] [A' B'] [f g].

  exact (functor_ab_tensor_prod f g).


(** The tensor product functor is a bifunctor. *)

Instance is1bifunctor_ab_tensor_prod : Is1Bifunctor ab_tensor_prod.

  rapply Build_Is1Bifunctor'.

  snapply Build_Is1Functor.

 intros AB A'B' fg f'g' [p q].

    exact (functor2_ab_tensor_prod p q).

    exact (functor_ab_tensor_prod_id A B).

 intros AA' BB' CC' [f g] [f' g'].

    exact (functor_ab_tensor_prod_compose f f' g g').


(** ** Symmetry of the Tensor Product *)

(** The tensor product is symmetric in that the order in which we take the tensor shouldn't matter up to isomorphism. *)

(** We can define a swap map which swaps the order of simple tensors. *)

Definition ab_tensor_swap {A B} : ab_tensor_prod A B $-> ab_tensor_prod B A.

  snapply ab_tensor_prod_rec.

 exact (flip tensor).

    apply tensor_dist_r.

    apply tensor_dist_l.


(** [ab_tensor_swap] is involutive. *)

Definition ab_tensor_swap_swap {A B}
  : ab_tensor_swap $o @ab_tensor_swap A B $== Id _.

  snapply ab_tensor_prod_ind_homotopy.

 

(** [ab_tensor_swap] is natural in both arguments. This means that it also acts on tensor functors. *)

Definition ab_tensor_swap_natural {A B A' B' : AbGroup} (f : A $-> A') (g : B $-> B')
  : ab_tensor_swap $o functor_ab_tensor_prod f g
    $== functor_ab_tensor_prod g f $o ab_tensor_swap.

  snapply ab_tensor_prod_ind_homotopy.

 (* This speeds up the [reflexivity] and the [Defined]. *)


(** The swap map gives us a symmetric braiding on the category of abelian groups. We will later show it is a full symmetric monoidal category. *)

Instance symmetricbraiding_ab_tensor_prod : SymmetricBraiding ab_tensor_prod.

  snapply Build_SymmetricBraiding.

 snapply Build_NatTrans.

 intro; exact ab_tensor_swap.

 snapply Build_Is1Natural.

      intros; napply ab_tensor_swap_natural.

 intros; napply ab_tensor_swap_swap.

 

(** ** Twisting Triple Tensors *)

(** In order to construct the symmetric monoidal category, we will use what is termed the "Twist construction" in Monoidal.v. This simplifies the data of a symmetric monoidal category by constructing it from simpler parts. For instance, instead of having to prove full associativity [(A ⊗ B) ⊗ C $-> A ⊗ (B ⊗ C)], we can provide a twist map [A ⊗ (B ⊗ C) $-> B ⊗ (A ⊗ C)] and use the symmetric braiding we have so far to prove associativity. *)

(** In order to be more efficient whilst unfolding definitions, we break up the definition of a twist map into its components. *)

Local Definition ab_tensor_prod_twist_map {A B C : AbGroup}
  : A -> (ab_tensor_prod B C $-> ab_tensor_prod B (ab_tensor_prod A C)).

  snapply ab_tensor_prod_rec'.

    exact (grp_homo_tensor_l b $o grp_homo_tensor_l a).

    napply tensor_dist_r.


Local Definition ab_tensor_prod_twist_map_additive_l {A B C : AbGroup}
  (a a' : A) (b : ab_tensor_prod B C)
  : ab_tensor_prod_twist_map (a + a') b
    = ab_tensor_prod_twist_map a b + ab_tensor_prod_twist_map a' b.

  napply ab_tensor_prod_ind_homotopy_plus.

  change (tensor b (tensor (a + a') c)
    = tensor b (tensor a c) + tensor b (tensor a' c)).

 
  rhs_V napply tensor_dist_l.

  napply (ap (tensor b)).

  napply tensor_dist_r.


(** Given a triple tensor product, we have a twist map which permutes the first two components. *)

Definition ab_tensor_prod_twist {A B C}
  : ab_tensor_prod A (ab_tensor_prod B C) $-> ab_tensor_prod B (ab_tensor_prod A C).

  snapply ab_tensor_prod_rec'.

 exact ab_tensor_prod_twist_map.

 exact ab_tensor_prod_twist_map_additive_l.


(** The twist map is involutive. *)

Definition ab_tensor_prod_twist_twist {A B C}
  : ab_tensor_prod_twist $o @ab_tensor_prod_twist A B C $== Id _.

  snapply ab_tensor_prod_ind_homotopy_triple.


(** The twist map is natural in all 3 arguments. This means that the twist map acts on the triple tensor functor in the same way. *)

Definition ab_tensor_prod_twist_natural {A B C A' B' C' : AbGroup}
  (f : A $-> A') (g : B $-> B') (h : C $-> C')
  : ab_tensor_prod_twist $o fmap11 ab_tensor_prod f (fmap11 ab_tensor_prod g h)
    $== fmap11 ab_tensor_prod g (fmap11 ab_tensor_prod f h) $o ab_tensor_prod_twist.

  snapply ab_tensor_prod_ind_homotopy_triple.

  (* This [change] speeds up the [reflexivity].  [simpl] produces a goal that looks the same, but is still slow. *)

change (tensor (g b) (tensor (f a) (h c)) = tensor (g b) (tensor (f a) (h c))).


(** ** Unitality of [abgroup_Z] *)

(** In the symmetric monoidal structure on abelian groups, [abgroup_Z] is the unit. We show that tensoring with [abgroup_Z] on the right is isomorphic to the original group. *)

(** First we characterise the action of integers via [grp_pow] and their interaction on tensors. This is just a generalisation of the distributivity laws for tensors. *)

(** Multiplication in the first factor can be factored out. *)

Definition tensor_ab_mul_l {A B : AbGroup} (z : Int) (a : A) (b : B)
  : tensor (ab_mul z a) b = ab_mul z (tensor a b)
  := ab_mul_natural (grp_homo_tensor_r b) z a.


(** Multiplication in the second factor can be factored out. *)

Definition tensor_ab_mul_r {A B : AbGroup} (z : Int) (a : A) (b : B)
  : tensor a (ab_mul z b) = ab_mul z (tensor a b)
  := ab_mul_natural (grp_homo_tensor_l a) z b.


(** Multiplication can be transferred from one factor to the other. The tensor product of [R]-modules will include this as an extra axiom, but here we have [Z]-modules and we can prove it. *)

Definition tensor_ab_mul {A B : AbGroup} (z : Int) (a : A) (b : B)
  : tensor (ab_mul z a) b = tensor a (ab_mul z b).

  rhs napply tensor_ab_mul_r.

  napply tensor_ab_mul_l.


(** [abgroup_Z] is a right identity for the tensor product. *)

Definition ab_tensor_prod_Z_r {A}
  : ab_tensor_prod A abgroup_Z $<~> A.

  (** Checking that the inverse map is a homomorphism is easier. *)

  snapply Build_GroupIsomorphism.

 napply grp_homo_tensor_r.

 snapply isequiv_adjointify.

 snapply ab_tensor_prod_rec'.

 intros a a' z; cbn beta.

        napply (grp_homo_op (ab_mul z)).

      change (forall x : ?A, (grp_homo_map ?f) ((grp_homo_map ?g) x) = x)
        with (f $o g $== Id _).

      snapply ab_tensor_prod_ind_homotopy.

      change (tensor (B:=abgroup_Z) (grp_pow a z) 1%int = tensor a z).

      lhs napply tensor_ab_mul.

      lhs napply abgroup_Z_ab_mul.


(** We have a right unitor for the tensor product given by unit [abgroup_Z]. Naturality of [ab_tensor_prod_Z_r] is straightforward to prove. *)

Instance rightunitor_ab_tensor_prod
  : RightUnitor ab_tensor_prod abgroup_Z.

  snapply Build_NatEquiv.

    apply ab_tensor_prod_Z_r.

 snapply Build_Is1Natural.

    snapply ab_tensor_prod_ind_homotopy.

    intros a z; symmetry.

    exact (grp_pow_natural _ _ _).


(** Since we have symmetry of the tensor product, we get left unitality for free. *)

Instance left_unitor_ab_tensor_prod
  : LeftUnitor ab_tensor_prod abgroup_Z.

  rapply left_unitor_twist.


(** ** Symmetric Monoidal Structure of Tensor Product *)

(** Using the twist construction we can derive an associator for the tensor product. In other words, we have associativity of the tensor product of abelian groups natural in each factor. *)

Instance associator_ab_tensor_prod : Associator ab_tensor_prod.

  srapply associator_twist.

 exact @ab_tensor_prod_twist.

 intros; napply ab_tensor_prod_twist_twist.

 intros; napply ab_tensor_prod_twist_natural.


(** The triangle identity is straightforward to prove using the custom induction principles we proved earlier. *)

Instance triangle_ab_tensor_prod
  : TriangleIdentity ab_tensor_prod abgroup_Z.

  snapply triangle_twist.

  snapply ab_tensor_prod_ind_homotopy_triple.

  intros a b z; symmetry.

  exact (tensor_ab_mul z a b).


(** The hexagon identity is also straightforward to prove. We simply have to reduce all the involved functions on the simple tensors using our custom triple tensor induction principle. *)

Instance hexagon_ab_tensor_prod : HexagonIdentity ab_tensor_prod.

  snapply hexagon_twist.

  snapply ab_tensor_prod_ind_homotopy_triple.

  change (tensor c (tensor a b) = tensor c (tensor a b)).


(** Finally, we can prove the pentagon identity using the quadruple tensor induction principle. As we did before, the work only involves reducing the involved functions on the simple tensor redexes. *)

Instance pentagon_ab_tensor_prod : PentagonIdentity ab_tensor_prod.

  snapply pentagon_twist.

  snapply ab_tensor_prod_ind_homotopy_quad.

  change (tensor c (tensor d (tensor a b)) = tensor c (tensor d (tensor a b))).


(** We therefore have all the data of a monoidal category. *)

Instance ismonoidal_ab_tensor_prod
  : IsMonoidal AbGroup ab_tensor_prod abgroup_Z
  := {}.


(** And furthermore, all the data of a symmetric monoidal category. *)

Instance issymmmetricmonoidal_ab_tensor_prod
  : IsSymmetricMonoidal AbGroup ab_tensor_prod abgroup_Z
  := {}.


(** ** Preservation of Coequalizers *)

(** The tensor product of abelian groups preserves coequalizers, meaning that the coequalizer of two tensored groups is the tensor of the coequalizer. We show this is the case on the left and the right. *)

(** Tensor products preserve coequalizers on the right. *)

Definition grp_iso_ab_tensor_prod_coeq_l A {B C : AbGroup} (f g : B $-> C)
  : ab_coeq (fmap01 ab_tensor_prod A f) (fmap01 ab_tensor_prod A g)
    $<~> ab_tensor_prod A (ab_coeq f g).

  snapply cate_adjointify.

 snapply ab_coeq_rec.

 rapply (fmap01 ab_tensor_prod A).

 refine (_^$ $@ fmap02 ab_tensor_prod _ _ $@ _).

      1,3: tapply fmap01_comp.

      napply ab_coeq_glue.

 snapply ab_tensor_prod_rec'.

      snapply functor_ab_coeq.

      1,2: snapply (grp_homo_tensor_l a).

      1,2: hnf; reflexivity.

 intros a a'; cbn beta.

      srapply ab_coeq_ind_hprop.

      exact (ap (ab_coeq_in
        (f:=fmap01 ab_tensor_prod A f)
        (g:=fmap01 ab_tensor_prod A g))
        (tensor_dist_r a a' x)).

 snapply ab_tensor_prod_ind_homotopy.

    srapply ab_coeq_ind_hprop.

 snapply ab_coeq_ind_homotopy.

    snapply ab_tensor_prod_ind_homotopy.


(** The equivalence respects the natural maps from [ab_tensor_prod A C]. *)

Definition ab_tensor_prod_coeq_l_triangle A {B C : AbGroup} (f g : B $-> C)
  : grp_iso_ab_tensor_prod_coeq_l A f g $o ab_coeq_in
    $== fmap01 ab_tensor_prod A ab_coeq_in.

  snapply ab_tensor_prod_ind_homotopy.


(** Tensor products preserve coequalizers on the left. *)

Definition grp_iso_ab_tensor_prod_coeq_r {A B : AbGroup} (f g : A $-> B) C
  : ab_coeq (fmap10 ab_tensor_prod f C) (fmap10 ab_tensor_prod g C)
    $<~> ab_tensor_prod (ab_coeq f g) C.

  refine (braide _ _ $oE _).

  nrefine (grp_iso_ab_tensor_prod_coeq_l _ f g $oE _).

  snapply grp_iso_ab_coeq.

  1,2: symmetry; napply ab_tensor_swap_natural.


(** The equivalence respects the natural maps from [ab_tensor_prod B C]. *)

Definition ab_tensor_prod_coeq_r_triangle {A B : AbGroup} (f g : A $-> B) C
  : grp_iso_ab_tensor_prod_coeq_r f g C $o ab_coeq_in
    $== fmap10 ab_tensor_prod ab_coeq_in C.

  snapply ab_tensor_prod_ind_homotopy.


(** ** Tensor Product of Free Abelian Groups *)

Definition equiv_ab_tensor_prod_freeabgroup X Y
  : FreeAbGroup (X * Y) $<~> ab_tensor_prod (FreeAbGroup X) (FreeAbGroup Y).

  srefine (let f:=_ in let g:=_ in cate_adjointify f g _ _).

 snapply FreeAbGroup_rec.

    exact (tensor (freeabgroup_in x) (freeabgroup_in y)).

 snapply ab_tensor_prod_rec.

      snapply FreeAbGroup_rec.

      snapply FreeAbGroup_rec.

      snapply grp_homo_op.

      rapply Abel_ind_hprop.

      snapply (FreeGroup_ind_homotopy _ (f' := sgop_hom _ _)).

      lhs napply FreeGroup_rec_beta.

      lhs napply grp_homo_op.

      snapply (ap011 (+) _^ _^).

      1,2: napply FreeGroup_rec_beta.

 snapply ab_tensor_prod_ind_homotopy.

    change (f $o g $o grp_homo_tensor_l x $== grp_homo_tensor_l x).

    rapply Abel_ind_hprop.

    change (@abel_in ?G) with (grp_homo_map (@abel_unit G)).

    repeat change (cat_comp (A:=AbGroup) ?f ?g) with (cat_comp (A:=Group) f g).

    change (forall y, grp_homo_map ?f (abel_unit y) = grp_homo_map ?g (abel_unit y))
      with (cat_comp (A:=Group) f abel_unit $== cat_comp (A:=Group) g abel_unit).

    rapply FreeGroup_ind_homotopy.

    change (f $o g $o grp_homo_tensor_r (freeabgroup_in y) $== grp_homo_tensor_r (freeabgroup_in y)).

    rapply Abel_ind_hprop.

    change (@abel_in ?G) with (grp_homo_map (@abel_unit G)).

    repeat change (cat_comp (A:=AbGroup) ?f ?g) with (cat_comp (A:=Group) f g).

    change (forall y, grp_homo_map ?f (abel_unit y) = grp_homo_map ?g (abel_unit y))
      with (cat_comp (A:=Group) f abel_unit $== cat_comp (A:=Group) g abel_unit).

    rapply FreeGroup_ind_homotopy.

 rapply Abel_ind_hprop.

    change (GpdHom (A:=Hom(A:=Group) (FreeGroup (X * Y)) _)
      (cat_comp (A:=Group) (g $o f) (@abel_unit (FreeGroup (X * Y))))
      (@abel_unit (FreeGroup (X * Y)))).

    snapply FreeGroup_ind_homotopy.


(** ** Tensor products distribute over direct sums *)

Definition ab_tensor_prod_dist_l {A B C : AbGroup}
  : ab_tensor_prod A (ab_biprod B C)
    $<~> ab_biprod (ab_tensor_prod A B) (ab_tensor_prod A C).

  stapply (let f := _ in let g := _ in cate_adjointify f g _ _).

 snapply ab_tensor_prod_rec.

      exact (tensor a (fst bc), tensor a (snd bc)).

 intros a bc bc'; cbn beta.

      snapply path_prod'; snapply tensor_dist_l.

 intros a a' bc; cbn beta.

      snapply path_prod; snapply tensor_dist_r.

 snapply ab_biprod_rec.

 exact (fmap01 ab_tensor_prod A ab_biprod_inl).

 exact (fmap01 ab_tensor_prod A ab_biprod_inr).

 snapply ab_biprod_ind_homotopy.

 refine (cat_assoc _ _ _ $@ (_ $@L _) $@ _).

 
      1: snapply ab_biprod_rec_beta_inl.

      snapply ab_tensor_prod_ind_homotopy.

      snapply path_prod; simpl.

 snapply tensor_zero_r.

 refine (cat_assoc _ _ _ $@ (_ $@L _) $@ _).

      1: snapply ab_biprod_rec_beta_inr.

      snapply ab_tensor_prod_ind_homotopy.

      snapply path_prod; simpl.

 snapply tensor_zero_r.

 snapply ab_tensor_prod_ind_homotopy.

    lhs_V napply tensor_dist_l; simpl.

    symmetry; apply grp_prod_decompose.


Definition ab_tensor_prod_dist_r {A B C : AbGroup}
  : ab_tensor_prod (ab_biprod A B) C
    $<~> ab_biprod (ab_tensor_prod A C) (ab_tensor_prod B C).

  refine (emap11 ab_biprod (braide _ _) (braide _ _)
    $oE _ $oE braide _ _).

  snapply ab_tensor_prod_dist_l.


(** TODO: Show that the category of abelian groups is symmetric closed and therefore we have adjoint pair with the tensor and internal hom. This should allow us to prove lemmas such as tensors distributing over coproducts. *)

Timings for TensorProduct.v