nLab indexed tensor product

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Context

Equivariant higher algebra

Higher algebra

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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.