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Orbital categories and weak indexing systems

Natalie Stewart

TL;DR

This work develops a comprehensive combinatorial framework for equivariant arities by introducing and studying the poset of weak ${\mathcal T}$-indexing systems ${\mathrm wIndex}_{\mathcal T}$ for an orbital category ${\mathcal T}$. It proves an equivalence with weak indexing categories and situates unital and almost unital variants within a network of adjunctions and fibrations (color-support, unit, transfer, fold, and essence) that organize how arities behave under restrictions, inductions, and coproducts. The paper then establishes closure and join operations, coinduction, and principal indexing structures, and develops explicit enumerative results, including a finitary classification for ${\mathcal O}_{C_{p^n}}$ and a detailed Cp-analysis, revealing both the richness and finiteness of the indexing data. Together with the transfer-fold framework, these results illuminate how ${\mathcal N}_\infty$-operads and their tensor products arise in equivariant higher algebra, with potential applications to global equivariant homotopy theory and Eckmann–Hilton phenomena. The framework thus provides a precise, combinatorial toolkit for identifying and classifying arities of equivariant multiplicative structures across a broad class of orbital categories.

Abstract

We initiate the combinatorial study of the poset $\mathrm{wIndex}_{\mathcal{T}}$ of weak $\mathcal{T}$-indexing systems, consisting of composable collections of arities for $\mathcal{T}$-equivariant algebraic structures, where $\mathcal{T}$ is an orbital $\infty$-category, such as the orbit category of a finite group. In particular, we show that these are equivalent to weak $\mathcal{T}$-indexing categories and characterize various unitality conditions. Within this sits a natural generalization $\mathrm{Index}_{\mathcal{T}} \subset \mathrm{wIndex}_{\mathcal{T}}$ of Blumberg-Hill's indexing systems, consisting of arities for structures possessing binary operations and unit elements. We characterize the relationship between the posets of unital weak indexing systems and indexing systems, the latter remaining isomorphic to transfer systems on this level of generality. We use this to characterize the poset of unital $C_{p^n}$-weak indexing systems.

Orbital categories and weak indexing systems

TL;DR

This work develops a comprehensive combinatorial framework for equivariant arities by introducing and studying the poset of weak -indexing systems for an orbital category . It proves an equivalence with weak indexing categories and situates unital and almost unital variants within a network of adjunctions and fibrations (color-support, unit, transfer, fold, and essence) that organize how arities behave under restrictions, inductions, and coproducts. The paper then establishes closure and join operations, coinduction, and principal indexing structures, and develops explicit enumerative results, including a finitary classification for and a detailed Cp-analysis, revealing both the richness and finiteness of the indexing data. Together with the transfer-fold framework, these results illuminate how -operads and their tensor products arise in equivariant higher algebra, with potential applications to global equivariant homotopy theory and Eckmann–Hilton phenomena. The framework thus provides a precise, combinatorial toolkit for identifying and classifying arities of equivariant multiplicative structures across a broad class of orbital categories.

Abstract

We initiate the combinatorial study of the poset of weak -indexing systems, consisting of composable collections of arities for -equivariant algebraic structures, where is an orbital -category, such as the orbit category of a finite group. In particular, we show that these are equivalent to weak -indexing categories and characterize various unitality conditions. Within this sits a natural generalization of Blumberg-Hill's indexing systems, consisting of arities for structures possessing binary operations and unit elements. We characterize the relationship between the posets of unital weak indexing systems and indexing systems, the latter remaining isomorphic to transfer systems on this level of generality. We use this to characterize the poset of unital -weak indexing systems.
Paper Structure (38 sections, 61 theorems, 133 equations, 2 figures)

This paper contains 38 sections, 61 theorems, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Orbital categories and indexed coproducts
  3. Orbital categories
  4. Indexed coproducts
  5. Weak indexing systems and weak indexing categories
  6. Weak indexing systems
  7. Some examples
  8. Weak indexing categories
  9. Unital weak indexing categories and transfer systems
  10. Why (unital) weak indexing systems?
  11. Notation and conventions
  12. Weak indexing systems
  13. Weak indexing categories vs weak indexing systems
  14. Reduction to the case with a terminal object
  15. The comparison theorem
  16. ...and 23 more sections

Key Result

Lemma 10

If ${\mathcal{T}}$ is (atomic) orbital and $V \in {\mathcal{T}}$ is an object, then ${\mathcal{T}}_{/V}$ is (atomic) orbital.

Figures (2)

  • Figure 1: Pictured is the result of Rubin and Balchin-Barnes-Rotzheim's computation of $\mathrm{Transf}_{C_{p^2}}$.
  • Figure 2: Pictured is a Hasse diagram for the poset of unital $C_{p^2}$-weak indexing systems. Dashed lines separate the fibers of the cocartesian fibration $({\mathfrak{R}},\nabla)$.

Theorems & Definitions (152)

  • example 3
  • definition 4: Nardin-Stable
  • example 5
  • example 6
  • definition 7
  • example 9
  • Lemma 10
  • proof
  • definition 11
  • example 12
  • ...and 142 more