Orbital categories and weak indexing systems

Natalie Stewart
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引用次数: 0

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.
轨道类别和弱索引系统
我们开始对弱$\mathcal{T}$indexing系统的poset$\mathrm{wIndex}_{\mathcal{T}}$进行组合研究,该系统由$\mathcal{T}$-后代数结构的可组合的arities集合组成,其中$\mathcal{T}$是一个轨道$\infty$范畴,如有限群的轨道范畴。特别是,我们证明了这些等价于弱$\mathcal{T}$索引范畴,并描述了各种单位性条件。在这个范畴中,有一个自然的广义范畴 $\mathrm{Index}_{\mathcal{T}}\subset \mathrm{wIndex}_{\mathcal{T}}$ 是布伦贝格-希尔索引系统的子集,由拥有二元运算和单元的结构的算术组成。我们描述了单元弱索引系统的posets 与索引系统之间的关系,后者在这个广义层次上仍然与转移系统同构。我们以此来描述单元$C_{p^n}$弱索引系统的集合。
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