Orbital categories and weak indexing systems

TL;DR

This paper develops a combinatorial framework for weak $ ext{T}$-indexing systems in the context of orbital $ ext{infinity}$-categories, generalizing existing indexing systems and characterizing their relationships and specific cases.

Contribution

It introduces the poset of weak $ ext{T}$-indexing systems, relates it to unital systems and transfer systems, and characterizes unital $C_{p^n}$-weak indexing systems.

Findings

Equivalence between weak $ ext{T}$-indexing systems and categories.

Characterization of unital weak indexing systems.

Description of unital $C_{p^n}$-weak indexing systems.

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.