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

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

2. (composition) for all $S \in \underline{\mathbb{F}}_I$, $\underline{\mathbb{F}}_I \subset \underline{\mathbb{F}}_{\mathcal{T}}$ is closed under $S$-indexed coproducts.

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

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

Indexing categories

Definition

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

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

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

3. ($\Sigma$-action) $I$ contains all automorphisms of its objects.

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

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

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

Theorem

The assignment $I \mapsto \underline{\mathbb{F}}_I$ 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 $\underline{\mathbb{F}}_I$ is almost essentially unital (or aE-unital) if for all non-contractible $S \in \mathbb{F}_{I,V}$, the empty $V$-set $\emptyset_V$ is in $\mathbb{F}_{I,V}$.

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

Transfer systems

Let $\mathrm{Sub}(G)$ be the subgroup lattice of $G$. We say that a subposet $R \subset \mathrm{Sub}(G)$ is a 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 2017 and by Balchin, Barnes & Roitzheim 2019.

Theorem

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

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

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 $\mathrm{Sub}(G) \rightarrow \mathcal{O}_G$. Thus we may make the following definition.

Definition

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

Note that closure under summands implies that, whenever $I$ is a unital weak indexing category, its intersection $\mathfrak{R}(I) \coloneqq I \cap \mathcal{T}$ 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 $\mathfrak{R}\colon \mathrm{wIndex}_{\mathcal{T}}^{\mathrm{uni}} \rightarrow \mathrm{Transf}_{\mathcal{T}}$ has a fully faithful right adjoint whose essential image is spanned by the $\mathcal{T}$-indexing systems.

Properties

$N_\infty$-operads.

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

(…)

Related concepts

• G-operad, N-∞ operad

• G-symmetric monoidal category

• G-semiadditivity

References

Original discussion:

• Andrew Blumberg, Michael Hill, Operadic multiplications in equivariant spectra, norms, and transfers, Advances in Mathematics 285 (2015) 658-708 [arXiv:1309.1750, doi:10.1016/j.aim.2015.07.013]

Further characterization:

• Andrew Blumberg, Michael Hill, Incomplete Tambara functors, Algebr. Geom. Topol. 18 (2018) 723-766 [arXiv:1603.03292, doi:10.2140/agt.2018.18.723]

• Jonathan Rubin, Combinatorial $N_\infty$ operads [arXiv:1705.03585v3]

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 $p^m q^n$, [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 $p^{n} q^{m}$ by linear isometries $N_\infty$-operads [arXiv:2311.01608]

On enumerating categories of transfer systems

• Scott Balchin, David Barnes, Constanze Roitzheim, $N_\infty$-operads and associahedra, Pacific J. Math. 315 (2021) 285-304 [arXiv:1905.03797, doi:10.2140/pjm.2021.315.285]

• Evan E. Franchere, Kyle Ormsby, Angélica Osorno, Weihang Qin, Riley Waugh, Self-duality of the lattice of transfer systems via weak factorization systems [arXiv:2102.04415]

• Scott Balchin, Ethan MacBrough, Kyle Ormsby, Lifting $N_\infty$ operads from conjugacy data, [arXiv:2209.06798]

• Scott Balchin, Ethan MacBrough, Kyle Ormsby, The combinatorics of $N_\infty$ operads for $C_{qp^n}$ and $D_{p^n}$, [arXiv:2209.06992]

Over orbital categories:

• Denis Nardin, Jay Shah, Parametrized and equivariant higher algebra [arxiv:2203.00072]

• Natalie Stewart, Orbital categories and weak indexing systems [arXiv:2409.01377]