nLab indexing system

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Idea

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

Definitions

Indexing systems

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.

Definition

A full 𝒯\mathcal{T}-subcategory 𝔽̲ I𝔽̲ 𝒯\underline{\mathbb{F}}_I \subset \underline{\mathbb{F}}_{\mathcal{T}} is called a weak indexing system if

  1. (objects) whenever the VV-value 𝔽 I,V\mathbb{F}_{I,V} is nonempty, it contains the VV-set * V*_V.

  2. (composition) for all S𝔽̲ IS \in \underline{\mathbb{F}}_I, 𝔽̲ I𝔽̲ 𝒯\underline{\mathbb{F}}_I \subset \underline{\mathbb{F}}_{\mathcal{T}} is closed under SS-indexed coproducts.

    𝔽̲ I\underline{\mathbb{F}}_I is called an indexing system if, additionally,

  3. (coproducts) for all nn \in \mathbb{N} and V𝒯V \in \mathcal{T}, n* V𝔽 I,Vn \cdot *_V \in \mathbb{F}_{I,V}

Indexing categories

Definition

A subcategory I𝔽 𝒯I \subset \mathbb{F}_{\mathcal{T}} is called a weak indexing category if it satisfies the following conditions:

  1. (restriction-stability) morphisms in II is stable under pullbacks along arbitrary maps in 𝔽 𝒯\mathbb{F}_{\mathcal{T}}.

  2. (Segal condition) A pair of maps TST \rightarrow S and TST' \rightarrow S' are in II if and only if their coproduct TTSST \sqcup T' \rightarrow S \sqcup S' is in II; and

  3. (Σ\Sigma-action) II contains all automorphisms of its objects.

A weak indexing category I𝔽 𝒯I \subset \mathbb{F}_{\mathcal{T}} is called an indexing category if it contains the fold map nVVn \cdot V \rightarrow V for all VcTV \in \cT and nn \in \mathbb{N}.

Given II a weak indexing category, we may define a full 𝒯\mathcal{T}-subcategory

(𝔽̲ I) V{SInd V 𝒯SVI}𝔽 V. (\underline{\mathbb{F}}_I)_V \coloneqq \{S \mid \Ind_V^{\mathcal{T}} S \rightarrow V \in I\} \subset \mathbb{F}_{V}.

Theorem

The assignment I𝔽̲ II \mapsto \underline{\mathbb{F}}_I furnishes an equivalence between the posets of weak indexing categories and weak indexing systems; this restricts to an equivalence between indexing categories and indexing systems.

Unitality conditions

Definition

  1. A weak indexing system 𝔽̲ I\underline{\mathbb{F}}_I is almost essentially unital (or aE-unital) if for all non-contractible S𝔽 I,VS \in \mathbb{F}_{I,V}, the empty VV-set V\emptyset_V is in 𝔽 I,V\mathbb{F}_{I,V}.

  2. A weak indexing system 𝔽̲ I\underline{\mathbb{F}}_I is unital if V𝔽 I,V\emptyset_V \in \mathbb{F}_{I,V} for all V𝒯V \in \mathcal{T}.

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 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 2017 and by Balchin, Barnes & Roitzheim 2019.

Theorem

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

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

It is not clear how to generalize the subgroup lattice to orbital \infty-categories. Instead, one should note that closure under conjugation implies that a transfer system only depends on its image under the homogeneous G-set functor Sub(G)𝒪 G\mathrm{Sub}(G) \rightarrow \mathcal{O}_G. Thus we may make the following definition.

Definition

A core-containing wide subcategory 𝒯 R𝒯\mathcal{T}^{\simeq} \subset R \subset \mathcal{T} is a \emph{transfer system} if, for all diagrams in 𝒯\mathcal{T} with α\alpha in RR and whose map UU× VVU' \rightarrow U \times_V V' is a summand inclusion, the map α\alpha' is in RR.

Note that closure under summands implies that, whenever II is a unital weak indexing category, its intersection (I)I𝒯\mathfrak{R}(I) \coloneqq I \cap \mathcal{T} is a transfer system. Nardin-Shah 22 generalized the correspondence between indexing systems and transfer systems to orbital \infty-categories, and Stewart 24 generalized this to weak indexing systems as the following.

Proposition

The monotone map :wIndex 𝒯 uniTransf 𝒯\mathfrak{R}\colon \mathrm{wIndex}_{\mathcal{T}}^{\mathrm{uni}} \rightarrow \mathrm{Transf}_{\mathcal{T}} has a fully faithful right adjoint whose essential image is spanned by the 𝒯\mathcal{T}-indexing systems.

Properties

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 there for details).

(…)

References

Original discussion:

Further characterization:

On those coming from equivariant linear isometries operads:

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

  • Usman Hafeez, Peter Marcus, Kyle Orbsby, Angélica Osorno, Saturated and linear isometric transfer systems for cyclic groups of order p mq np^m q^n, [arXiv:2109.08210]

  • Ethan MacBrough, Equivariant linear isometries operads over Abelian groups, [arXiv:2311.08797]

  • Julie E. M. Bannwart, Realization of saturated transfer systems on cyclic groups of order p nq mp^{n} q^{m} by linear isometries N N_\infty-operads [arXiv:2311.01608]

On enumerating categories of transfer systems

Over orbital categories:

Last revised on September 5, 2024 at 19:00:43. See the history of this page for a list of all contributions to it.