nLab indexed tensor product

Redirected from "indexed products".
Note: indexed coproduct and indexed tensor product both redirect for "indexed products".
Contents

Context

Equivariant higher algebra

Higher algebra

Contents

Idea

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

Definition

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

Postcomposition along ψ S\psi_S yields a natural transformation between evaluation functors ev S=() S= [H/K i]Orb(S)() K i() H=ev H\ev_S = (-)_S = \prod_{[H/K_i] \in \mathrm{Orb}(S)} (-)_{K_i} \implies (-)_H = \ev_H, each sending Fun(Span(𝔽 G),Cat)\mathrm{Fun}(\mathrm{Span}(\mathbb{F}_G), \mathrm{Cat}). If 𝒞 Fun(Span(𝔽 G),Cat )\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 𝒞 \mathcal{C}^{\otimes} as the indexed tensor functor, and write it as

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

References

Originally,

Since then,

Created on July 30, 2024 at 14:07:56. See the history of this page for a list of all contributions to it.