nLab tensor product of algebras over a commutative monad



Higher algebra

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monoidal categories

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

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

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


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

or that

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

Moreover, if \otimes is symmetric, T\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)(A,a), (B,b)(B,b) and (R,r)(R,r) be TT-algebras. A binary morphism of algebras, or TT-balanced map, or TT-bilinear (or bimorphism, see the caveat above) from AA and BB to RR is a morphism f:ABRf:A\otimes B\to R of CC such that the following diagram commutes.

For nn 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.


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

We can denote by C T(A,B;C)C^T(A,B;C) the set of binary morphisms from AA and BB to CC this is a functor (C T) op×(C T) op×C TSet(C^T)^{op}\times(C^T)^{op}\times C^T\to Set. Analogous assignments can be given for nn inputs; this equips C TC^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.


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

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


See also


  • 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)

Last revised on April 26, 2024 at 10:02:36. See the history of this page for a list of all contributions to it.