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 Fuentas-Keuthan.

References

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