nLab commutative monad

Idea

The idea of a commutative monad could be motivated as either

• The monad equivalent of a commutative algebraic theory, generalized to either non-finitary theories or to categories other than Set. Intuitively, the operations involved have a commutativity property analogous to the one that monoids can have;
• A monad on a monoidal category whose underlying functor is lax monoidal, and whose unit and multiplication are monoidal. The interpretation is that the monad plays well with the monoidal product of the category, encoding for example products of formal expressions;
• A monad on a monoidal category whose left-strength and right-strength commute with each other.

It turns out that all these concepts coincide (see below, as well as the articles on commutative algebraic theories and monoidal monads).

Definition

Let $(T,\mu,\eta)$ be a monad on a symmetric monoidal category $C$, equipped with a left-strength $\sigma$ (and with the right-strength $\tau$ induced from $\sigma$ and the braiding).

We say that $T$ (or $\sigma$) is commutative if the following diagram commutes for all objects $X$ and $Y$ of $C$.

Note that this definition makes sense more generally in a monoidal category, for a monad equipped with a (non-necessarily symmetric) strength.

Properties

Relation with monoidal monads

The structure of a commutative monad is closely related to that of a monoidal monad, namely a monad in the bicategory of monoidal categories, lax monoidal functors and monoidal natural transformations.

For the details see this Proposition at monoidal monad.

On closed and monoidal closed categories

One can define left-strength and right-strength on a closed category, in a way that’s analogous to the definition in a monoidal category, and on monoidal closed categories the two definitions coincide (see strong monad - on closed and monoidal closed categories). The same is true for the notion of commutativity: one can define commutative monads on closed categories, and on closed monoidal categories the definition coincides with the definition given in this article.

More in detail, let $C$ be a closed category, and denote its internal homs by $[X,Y]$. Let $t':[X,Y]\to [T X, T Y]$ be a left-strength for $T$ (as defined here), and let $s':T[X,Y]\to [X, T Y]$ be a right-strength (as defined here). We say that $t'$ and $s'$ commute, or that $T$ is commutative, if the following diagram commutes. It can be shown by currying that on a monoidal closed category, this condition is equivalent to the “monoidal” definition of commutativity. A reference for this is Kock ‘71, Section 2.

Tensor product of algebras and multimorphisms

In many situations, algebras over a commutative monad are canonically equipped with a tensor product analogous to the tensor product of modules over a ring. In particular, commutative monads come equipped with a notion of multimorphisms of algebras, analogous to bilinear and multilinear maps.

On a closed category, analogously, the category of algebras inherits an internal hom, just like the one of modules. If the category is monoidal closed, the tensor product and the internal hom are adjoint to each other, making the category of algebra itself monoidal closed. This, once again, generalizes the hom-tensor adjunction of modules and vector spaces.

For the details, see tensor product of algebras over a commutative monad, and the section on the universal property for multimorphisms, as well as internal hom of algebras over a commutative monad.

Examples

In many of the examples, one can see how the property of commutativity (as opposed to just the structure of a strength) has really the meaning of the operations being commutative.

• The free commutative monoid monad on Set is commutative, the free monoid monad is not.
• If $M$ is a monoid, the action monad $X\mapsto M\times X$ on Set is commutative if and only if $M$ is commutative as a monoid.
• More generally, in an arbitrary symmetric monoidal category, the action monad induced by tensoring with an internal monoid is commutative if and only the monoid object is commutative. (See for example Brandenburg, Example 6.3.12.)

Since commutative monads are the same as monoidal monads, monads encoding structures which admit products are commutative:

• Most probability monads are commutative, with monoidal structure given by forming the product probability.
• The power set monad is commutative, with monoidal structure given by forming the product of subsets.

For a concrete example, consider the free commutative monoid monad $T$, and given a set $X$, write the elements of $TX$ as formal sums, such as $x_1+x_2$.
The commutativity condition says that these two compositions have the same result, and Note that these two expressions are not the same on the nose, they are the same because addition is commutative.

(See also the explanations at commutative algebraic theory.)

See also

• monoidal monad, commutative algebraic theory, strong monad, tensorial strength
• tensor product of algebras over a commutative monad
• internal hom of algebras over a commutative monad

References

• Anders Kock, Monads on symmetric monoidal closed categories, Arch. Math. 21 (1970) 1-10 [doi:10.1007/BF01220868]

• Anders Kock, Closed categories generated by commutative monads, Journal of the Australian Mathematical Society 12 4 (1971) 405-424 [doi:10.1017/S1446788700010272, pdf]

• Anders Kock, Strong functors and monoidal monads, Arch. Math 23 (1972) 113–120 [doi:10.1007/BF01304852, pdf]

• H. Lindner, Commutative monads in Deuxiéme colloque sur l’algébre des catégories. Amiens-1975. Résumés des conférences, pages 283-288. Cahiers de topologie et géométrie différentielle catégoriques, tome 16, nr. 3, 1975.

• 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)

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

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

• Paolo Perrone, Starting Category Theory, World Scientific, 2024, Chapter 6. (website)