nLab multiactegory

Idea

Just as a monoid can act on a set, a monoidal category can act on a category (such a structure is often called an actegory). A multiactegory is the appropriate generalisation from monoidal categories to multicategories.

Given a multicategory $M$, a (left) $M$-multiactegory $A$ has a collection of objects, and for each list $m_1, \ldots, m_n$ of objects of $M$ and each pair of objects $a, a' \in A$, a collection of multimorphisms

$$m_1, \ldots, m_n, a \to a'$$

along with composition operations that cohere with the composition and identities in $M$.

Representability

Just as monoidal categories can be identified with representable multicategories, actegories can be identified with representable multiactegories (see §3.4 of SZ24).

Related Concepts

• actegory

• multicategory, representable multicategory

References

Multiactegories were introduced in Definition 6.9 of the following paper, in the additional generality of skew-multicategories?. Ordinary multiactegories are skew-multicategories that are left- and right-normal in the terminology ibid.

• Nathanael Arkor, Dylan McDermott, The formal theory of relative monads, Journal of Pure and Applied Algebra 107676. (2024) [arXiv:2302.14014&rbrack.

The theory of representability for multiactegories is developed in §3.4 of:

• Mateusz Stroiński and Tony Zorman, Reconstruction of module categories in the infinite and non-rigid settings, arXiv:2409.00793 (2024).