symmetric monoidal (∞,1)-category of spectra
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
The idea of a commutative monad could be motivated as either
It turns out that all these concepts coincide (see below, as well as the articles on commutative algebraic theories and monoidal monads).
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.
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.
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.
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.
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.
Since commutative monads are the same as monoidal monads, monads encoding structures which admit products are commutative:
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.)
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)
Last revised on July 8, 2024 at 11:06:31. See the history of this page for a list of all contributions to it.