nLab indexing system

Idea

An indexing system is a combinatorial datum which uniquely determines an N-∞ operad.

Definition

In the following definition, fix $\mathcal{T}$ an orbital ∞-category and $\mathbb{F}_{\mathcal{T}} \coloneqq \mathcal{T}^{\amalg}$ its finite-coproduct closure; for instance, $\mathcal{T}$ may be the orbit category of a finite group, in which case $\mathbb{F}_{\mathcal{O}_G}$ is the category of finite G-sets.

Definition

A subcategory $I \subset \mathbb{F}_{\mathcal{T}}$ is called an indexing system if

1. ($\Sigma$-action) $I$ contains the core $\mathbb{F}_{\mathcal{T}}^{\simeq}$.

2. (Segal condition and restrictions) $I$ is stable under binary coproducts and pullbacks along arbitrary morphisms.

3. (Binary multiplications) $I$ contains the fold map $\nabla:2 \cdot V \rightarrow V$ for all $V \in \mathcal{T}$.

Equivalent characterizations

Sub-symmetric monoidal categories

Recall that induction yields an equivalence $\mathbb{F}_H \simeq \mathbb{F}_{G, /[G/H]}$ for each subgroup $H \subset G$. Given $I \subset \mathbb{F}_G$ an indexing system, and $H \subset G$ we refer to the corresponding subcategory

$$\mathbb{F}_{I,H} \coloneqq I_{/[G/H]} \simeq \mathbb{F}_{G, /[G/H]} \simeq \mathbb{F}_H.$$

The following was proved in Blumberg-Hill 16.

Theorem

There is a unique $G$-subcategory $\underline{\mathbb{F}}_I \subset \underline{\mathbb{F}}_G$; furthermore, this outlines an equivalence between the poset of indexing systems and the poset of full $G$-subcategories which contain trivial $H$-sets and are closed under coproducts, finite limits, and self-induction.

Transfer systems

Let $\mathrm{Sub}(G)$ be the subgroup lattice of $G$. We say that a subposet $R \subset \mathrm{Sub}(G)$ is a \emph{transfer system} if it is closed under conjugation and restriction.

Given $I \subset \mathbb{F}_G$ an indexing system, we let $T(I) \subset \mathrm{Sub}(G)$ denote the subposet consisting of inclusions $K \subset H$ such that the corresponding map $G/K \rightarrow G/H$ is in $I$. The following theorem was independently proved by Rubin and Balchin-Barnes-Roitzheim.

Theorem

$T(I)$ is a transfer system, and this outlines an equivalence of posets

$$\mathrm{Index}_{G} \simeq \mathrm{Transfer}_G.$$

$N_\infty$-operads.

The poset of subcommutative G-∞ operads containing $\mathbb{E}_\infty$ corresponds with $\mathrm{Index}_G$; these are called N-∞ operads (see the linked page for details).

Properties

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Related concepts

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References

Originally,

• Andrew Blumberg, Michael Hill, Operadic multiplications in equivariant spectra, norms, and transfers, (arXiv:1309.1750)

Further characterization,

• Andrew Blumberg, Michael Hill, Incomplete Tambara functors, (arXiv:1603.03292)

• Jonathan Rubin, Characterizations of equivariant Steiner and linear isometries operads, (arXiv:1903.08723v2)

• Scott Balchin, David Barnes, Constanze Roitzheim, $N_\infty$-operads and associahedra, (arXiv:1905.03797)

Over orbital categories,

• Denis Nardin, Jay Shah, Parametrized and equivariant higher algebra, (2022) (arxiv:2203.00072)