nLab indexing system




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


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


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

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

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

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

Equivalent characterizations

Sub-symmetric monoidal categories

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

𝔽 I,HI /[G/H]𝔽 G,/[G/H]𝔽 H. \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.


There is a unique GG-subcategory 𝔽̲ I𝔽̲ G\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 GG-subcategories which contain trivial HH-sets and are closed under coproducts, finite limits, and self-induction.

Transfer systems

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

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


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

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

N N_\infty-operads.

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






Further characterization,

Over orbital categories,

Last revised on May 12, 2024 at 00:32:03. See the history of this page for a list of all contributions to it.