Orbital categories and weak indexing systems

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.