标题： 高Segal空间和部分群 显示英文标题

标题： Higher Segal spaces and partial groups

摘要： Dyckerhoff 和 Kapranov 的$d$-Segal 条件是基于在$d$-维欧几里得空间中的循环多面体几何的单纯对象的精确性性质。 2-Segal 空间也被称为分解空间，大部分研究都集中在这种情况上。 我们研究这些条件与 Chermak 的部分群之间的相互作用，这是一个对称单纯集合的类。 $d$-Segal 条件对于对称单纯对象来说更为简化，并且在部分群的情况下具有特别明确的形式。 我们通过系统研究部分群$X$的“度”，即最小的$k\geq 1$使得$X$是$2k$-Segal，表明部分群提供了一个丰富的$d$-Segal 集类对于$d > 2$。 我们开发了有效的工具，根据部分群作用的离散几何来显式计算度，这是我们定义和研究的。 应用这些工具涉及解决抽象闭包空间的 Helly 类型问题。 我们在具体环境中进行度计算，包括此处引入的穿孔 Weyl 群，在这些情况下，我们发现度与相关半单李代数的阿贝尔子代数的最大维数密切相关。 显示英文摘要

摘要： The $d$-Segal conditions of Dyckerhoff and Kapranov are exactness properties for simplicial objects based on the geometry of cyclic polytopes in $d$-dimensional Euclidean space. 2-Segal spaces are also known as decomposition spaces, and most activity has focused on this case. We study the interplay of these conditions with the partial groups of Chermak, a class of symmetric simplicial sets. The $d$-Segal conditions simplify for symmetric simplicial objects, and take a particularly explicit form for partial groups. We show partial groups provide a rich class of $d$-Segal sets for $d > 2$, by undertaking a systematic study of the "degree" of a partial group $X$, namely the smallest $k\geq 1$ such that $X$ is $2k$-Segal. We develop effective tools to explicitly compute the degree based on the discrete geometry of actions of partial groups, which we define and study. Applying these tools involves solving Helly-type problems for abstract closure spaces. We carry out degree computations in concrete settings, including for the punctured Weyl groups introduced here, where we find that the degree is closely related to the maximal dimension of an abelian subalgebra of the associated semisimple Lie algebra.