nLab tensor product of algebras over a commutative monad

Idea

Algebras over a commutative monad, in many cases, admit a tensor product analogous to the traditional tensor product of modules over a ring.

This tensor product satisfies a universal property analogous to the one of the tensor product of modules, namely it represents multimorphisms in the same way in which the tensor product of modules represents bilinear maps.

Construction of the tensor product

Let $(T,\mu,\eta)$ be a commutative monad on a symmetric monoidal category $(C,\otimes,1)$. Denote the monoidal multiplication of $T$ by $\nabla$.

Given $T$-algebras $(A,a)$ and $(B,b)$, their tensor product is, if it exists, the object $A\otimes_T B$ given by the coequalizer in the Eilenberg-Moore category $C^T$

We say that $C^T$ has tensors if such equalizers exist for all $(A,a)$ and $(B,b)$. In that case, $\otimes_T$ is a functor $C^T\times C^T\to C^T$.

Theorem

(Seal ‘12, Corollary 2.5.6 and Theorem 2.6.4) Suppose that either:

• $C^T$ has tensors, and $\otimes_T$ preserves reflexive coequalizers in both variables separately (hence jointly);

or that

• $C$ has reflexive coequalizers, and that the functor $T(-\otimes -)$ preserves them in both variables separately (hence jointly).

Then $\otimes_T$ makes the Eilenberg-Moore category $C^T$ a monoidal category, with

• unit object given by $T1$, where $1$ is the unit object of $C$;
• unitors and associators induced by those of $\otimes$.

Moreover, if $\otimes$ is symmetric, $\otimes_T$ is symmetric too, with braiding induced by the one of $\otimes$.

Bimorphisms and universal property

Commutative monads admit a notion of multimorphism of algebras analogous to the notion of bilinear and multilinear map.

Caveat. Some authors call the analogue of a bilinear map a bimorphism. This is a distinct notion from “a morphism which is both epi and mono”, which some authors call by the same name. A less ambiguous term is binary morphism.

Let $(A,a)$, $(B,b)$ and $(R,r)$ be $T$-algebras. A binary morphism of algebras, or $T$-balanced map, or $T$-bilinear (or bimorphism, see the caveat above) from $A$ and $B$ to $R$ is a morphism $f:A\otimes B\to R$ of $C$ such that the following diagram commutes.

For $n$ inputs, we can define a multimorphism of algebras in the same way.

We can view a binary morphism as a morphism which is a “morphism of algebras in each variable separately”. This intuition can be made precise as follows.

Proposition

(Kock ‘71, Theorem 1.1, Seal ‘12, Proposition 2.1.2) A morphism $f:A\otimes B\to R$ of $C$ is a binary morphism of algebras if and only if both the following two diagrams commute. (The maps $s$ and $t$ denote the strength and costrength of the monad $T$.)

We can denote by $C^T(A,B;C)$ the set of binary morphisms from $A$ and $B$ to $C$ this is a functor $(C^T)^{op}\times(C^T)^{op}\times C^T\to Set$. Analogous assignments can be given for $n$ inputs; this equips $C^T$ with the structure of a multicategory extending its usual category structure. The most prominent example of this is multilinear maps extending linear maps.

As in the case of multilinear maps, we have that the tensor product of algebras, if it exists, represent multimorphisms.

Theorem

(Seal ‘12, Proposition 2.3.4) Suppose that $C^T$ has tensors. For each $T$-algebras $(A,a)$, $(B,b)$ and $(R,r)$ there is a bijection natural in $A$, $B$ and $C$.

In other words, $C^T$ with $\otimes_R$ is a representable multicategory.

The proof (see the reference above) follows closely the classical case of modules over a ring.

For monoidal closed categories

The treatment of the closed case goes back to the work done in the 70s by Anders Kock. A treatment of that case has been recently given also in the thesis of Martin Brandenburg.

If $C$ is a monoidal closed category and has equalizers, and $C^T$ has tensors, the first hypothesis of the theorem above is satisfied, and so $(C^T,\otimes_T,T1)$ is a monoidal category. Moreover, $C^T$ can be equipped with an internal hom $[A,B]_T$ which makes $C^T$ closed (see internal hom of algebras over a commutative monad). In that case the natural bijection given by the hom-tensor adjunction of $C$ induces a natural bijection which makes $C^T$ itself a closed monoidal category.

This generalizes the hom-tensor adjunction of modules, abelian groups and vector spaces.

See also closed monoidal structure on algebras over a commutative algebraic theory.

Examples

• tensor product of modules
• tensor product of abelian groups
• tensor product of algebras

See also

• monoidal monad, commutative monad, strong monad, commutative algebraic theory
• tensor product, tensor product of modules, multilinear map, representable multicategory
• internal hom of algebras over a commutative monad
• monoidal category, closed category, closed monoidal category

References

• Martin Brandenburg, Tensor categorical foundations of algebraic geometry (arXiv:1410.1716)

• William Keigher, Symmetric monoidal closed categories generated by commutative adjoint monads, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 19 no. 3 (1978), p. 269-293 (NUMDAM, pdf)

• Anders Kock, Monads on symmetric monoidal closed categories, Arch. Math. 21 (1970), 1–10.

• Anders Kock, Strong functors and monoidal monads, Arhus Universitet, Various Publications Series No. 11 (1970). PDF.

• Anders Kock, Closed categories generated by commutative monads, 1971 (pdf)

• Anders Kock, Bilinearity and cartesian closed monads, Mathematica Scandinavica, 29, 1971.

• Gavin J. Seal, Tensors, monads and actions (arXiv:1205.0101)