nLab symmetric multicategory

Idea

A symmetric multicategory is a multicategory equipped with an action of the symmetric group $S_n$ on the set of $n$-ary operations, for all $n$, such that composition respects this action.

Symmetric multicategories are equivalently called coloured symmetric operads over Set. See there for more details.

Properties

Closed monoidal structure

With respect to the Boardman-Vogt tensor product (see there for details) symmetric multicategories form a closed symmetric monoidal category.

Thomason model structure

A full subcategory of based symmetric multicategories admit a Thomason-style model structure that is Quillen equivalent to connective spectra, according to Fuentes-Keuthan.

Variations

An alternative “more unbiased” notion of symmetric multicategory uses families of objects indexed by arbitrary finite sets as the domains of morphisms, rather than lists indexed by standard ordered finite sets. Such definitions can be found in:

• Beilinson and Drinfeld 2004, under the name “pseudo-tensor categories”.
• Leinster 2004, Appendix A under the name “fat symmetric multicategories”.
• Pisani 2025 under the name “unbiased symmetric multicategories”.

References

• Daniel Fuentes-Keuthan?, Modeling connective spectra via multicategories, arXiv:1909.11148

• Alexander Beilinson, Vladimir Drinfeld, Chiral Algebras, Colloqium Publications 51, Amer. Math. Soc. 2004, gbooks

• Tom Leinster, Higher Operads, Higher Categories, Cambridge University Press 2004, arxiv

• Claudio Pisani, “Unbiased multicategory theory”, Theory and Applications of Categories, Vol. 44, 2025, No. 28, pp 826-868. web