nLab equivariant symmetric monoidal category

Idea

An equivariant symmetric monoidal category (Hill-Hopkins 16) is like a symmetric monoidal category but with the symmetric monoidal tensor product generalized to symmetric monoidal powers indexed by finite G-sets, for some group $G$.

Motivating applications come from equivariant homotopy theory.

Definition in terms of $\infty$-categories

Definition

If $\mathcal{T}$ is an orbital ∞-category and $\mathrm{Span}(\mathbb{F}_{\mathcal{T}})$ its Burnside category, then the $\infty$-category of small $\mathcal{T}$-symmetric monoidal $\infty$-categories is

$$\mathrm{Cat}_{\mathcal{T}}^{\otimes} := \mathrm{Fun}^{\times}(\mathrm{Span}(\mathbb{F}_{\mathcal{T}}), \mathrm{Cat}_{\infty}).$$

More generally, if $I \subset \mathbb{F}_{\mathcal{T}}$ is a weak indexing category, then $(\mathbb{F}_{\mathcal{T}}, \mathbb{F}_{\mathcal{T}}, I) \subset (\mathbb{F}_{\mathcal{T}}, \mathbb{F}_{\mathcal{T}}, \mathbb{F}_{\mathcal{T}})$ is a sub-algebraic triple in the sense of Barwick, so there is a Burnside category $\mathrm{Span}_I(\mathbb{F}_{\mathcal{T}}) \subset \mathrm{Span}(\mathbb{F}_{\mathcal{T}})$ whose forward maps are in $I$, and we define the $\infty$-category of small $I$-symmetric monoidal $\infty$-categories to be

$$\mathrm{Cat}_{I}^{\otimes} := \mathrm{Fun}^{\times}(\mathrm{Span}_I(\mathbb{F}_{\mathcal{T}}), \mathrm{Cat}_{\infty}).$$

In particular, $\mathcal{T}$-symmetric monoidal $\infty$-categories are simply $\mathcal{T}$-commutative monoids in $\mathrm{Cat}_\infty$.

Remark

In the case $\mathcal{T} = \mathcal{O}_G$ is the orbit category of a finite group, these are called $G$-symmetric monoidal $\infty$-categories, which is a source of potential confusion, as they are homotopical lifts of the symmetric monoidal Mackey functors considered in (Hill-Hopkins 16).

Related concepts

• G-commutative monoid

• G-operad

• Parametrized Higher Category Theory and Higher Algebra

References

• Clark Barwick, Spectral Mackey functors and equivariant algebraic K-theory (I), Adv. Math., 304:646–727, 2017 (doi:10.1016/j.aim.2016.08.043, arXiv:1404.0108)

• Michael Hill, Michael Hopkins, Equivariant symmetric monoidal structures (arXiv:1610.03114)

• Denis Nardin, Jay Shah, Parametrized and equivariant higher algebra, (2022) (arxiv:2203.00072)