nLab indexed tensor product

Idea

Indexed coproducts are to tensor products as equivariant symmetric monoidal categories are to symmetric monoidal categories.

Definition

Let $H \subset G$ be a subgroup of a finite group and $S \in \mathbb{F}_H$ a finite $H$-set. The equivalence $\Ind_H^G:\mathbb{F}_H \xrightarrow \sim \mathbb{F}_{G,/[G/H]}$ identifies $S$ with a canonical map $\psi_S:\Ind_H^G S \rightarrow [G/H]$.

Postcomposition along $\psi_S$ yields a natural transformation between evaluation functors $\ev_S = (-)_S = \prod_{[H/K_i] \in \mathrm{Orb}(S)} (-)_{K_i} \implies (-)_H = \ev_H$, each sending $\mathrm{Fun}(\mathrm{Span}(\mathbb{F}_G), \mathrm{Cat})$. If $\mathcal{C}^{\otimes} \in \mathrm{Fun}(\mathrm{Span}(\mathbb{F}_G), \mathrm{Cat}_\infty)$ is product-preserving (i.e. it is a G-symmetric monoidal ∞-category), then we refer to the value of this natural transformation on $\mathcal{C}^{\otimes}$ as the indexed tensor functor, and write it as

$$\bigotimes_{K_i}^S:\mathcal{C}_S = \prod_{H/K_i \in \mathrm{Orb}(S)} \mathcal{C}_{K_i} \rightarrow \mathcal{C}_H.$$

Related concepts

• Parametrized Higher Category Theory and Higher Algebra, G-∞-category, equivariant symmetric monoidal category

• equivariant homotopy theory

References

Originally,

• Michael Hill, Michael Hopkins, Douglas Ravenel, On the non-existence of elements of Kervaire invariant one, Annals of Mathematics 184 1 (2016)[doi:10.4007/annals.2016.184.1.1, arXiv:0908.3724, talk slides]

Since then,

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

• Shaul Barkan, Rune Haugseng, Jan Steinebrunner, Envelopes for Algebraic Patterns, (2022) (arXiv:2208.07183)