Higher Segal Spaces
Higher Segal Spaces
复制标题
更高的 Segal 空间
DOI：
10.1007/978-3-030-27124-4
复制
发表时间：
2012
影响因子：
--
通讯作者：
M. Kapranov
中科院分区：
文献类型：
--
作者：
Tobias Dyckerhoff;M. Kapranov
这是关于新的更高范畴结构的系列文章中的第一篇，该结构被称为更高的西格尔空间。对于任意的dc1，我们引入了d-西格尔空间的概念，d-西格尔空间是满足与d维循环多面体三角剖分有关的局部性条件的单纯空间.当d=1时，我们恢复了Rezk关于西格尔空间的理论.本文主要研究2-西格尔空间。该理论的出发点是观察到，正如前面所研究的那样，霍尔代数仅仅是一个更丰富的结构的影子，该结构由一个在2-西格尔空间的基准中捕获的更高相干系统所支配。这个2-西格尔空间是由Waldhausen的S构造给出的，它是代数K-理论中常见的单纯空间。2-西格尔空间的其他例子自然地出现在经典主题中，例如Hecke代数、循环杆结构、标志的配置空间、五角形方程的解和映射类群。
This is the first paper in a series on new higher categorical structures called higher Segal spaces. For every d C 1, we introduce the notion of a d-Segal space which is a simplicial space satisfying locality conditions related to triangulations of cyclic polytopes of dimension d. In the case d = 1, we recover Rezk’s theory of Segal spaces. The present paper focuses on 2-Segal spaces. The starting point of the theory is the observation that Hall algebras, as previously studied, are only the shadow of a much richer structure governed by a system of higher coherences captured in the datum of a 2-Segal space. This 2-Segal space is given by Waldhausen’s S-construction, a simplicial space familiar in algebraic K-theory. Other examples of 2-Segal spaces arise naturally in classical topics such as Hecke algebras, cyclic bar constructions, configuration spaces of flags, solutions of the pentagon equation, and mapping class groups.