nLab equivariant symmetric monoidal category

Redirected from "G-symmetric monoidal ∞-category".
Content

Context

Equivariant higher algebra

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

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 GG.

Motivating applications come from equivariant homotopy theory.

Definition in terms of \infty-categories

Definition

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

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

More generally, if I𝔽 𝒯I \subset \mathbb{F}_{\mathcal{T}} is a weak indexing category, then (𝔽 𝒯,𝔽 𝒯,I)(𝔽 𝒯,𝔽 𝒯,𝔽 𝒯)(\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 Span I(𝔽 𝒯)Span(𝔽 𝒯)\mathrm{Span}_I(\mathbb{F}_{\mathcal{T}}) \subset \mathrm{Span}(\mathbb{F}_{\mathcal{T}}) whose forward maps are in II, and we define the \infty-category of small II-symmetric monoidal \infty-categories to be

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

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

Remark

In the case 𝒯=𝒪 G\mathcal{T} = \mathcal{O}_G is the orbit category of a finite group, these are called GG-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 July 23, 2024 at 03:01:01. See the history of this page for a list of all contributions to it.