# nLab equivariant symmetric monoidal category

Content

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

#### Representation theory

representation theory

geometric representation theory

# Content

## 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

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

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).

## References

Last revised on April 21, 2024 at 16:52:34. See the history of this page for a list of all contributions to it.