无穷小和$B\_\infty$ -代数、有限空间和拟对称函数
Infinitesimal and $B\_\infty$-algebras, finite spaces, and quasi-symmetric functions
摘要：
有限拓扑空间与有限集上的预序是双射对应的。我们进行他们的研究，使用组合工具，已开发的一般离散结构的调查。一个特别的重点将放在最近的拓扑和组合霍普夫代数技术。我们将证明有限空间的线性跨度具有与拓扑学中常见的构造和结构密切相关的广义霍普夫代数结构（例如在闭路空间的研究中出现的结合代数范畴中的余群）。最引人注目的结果，我们得到的肯定是，有限空间的线性跨度进行结构的包络代数的$B\_\infty$ -代数，并有自然（Hopf代数）之间的有限空间和准对称函数的态射。在这个过程中，我们引入了舒尔-外尔范畴的概念，以描述刚性定理的范畴内的结合代数和相关结构的余群，以及说明存在的自然操作（分次置换）。
英文摘要：
Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be put on recent topological and combinatorial Hopf algebra techniques. We will show that the linear span of finite spaces carries generalized Hopf algebraic structures that are closely connected with familiar constructions and structures in topology (such as the one of cogroups in the category of associative algebras that has appeared e.g. in the study of loop spaces of suspensions). The most striking results that we obtain are certainly that the linear span of finite spaces carries the structure of the enveloping algebra of a $B\_\infty$--algebra, and that there are natural (Hopf algebraic) morphisms between finite spaces and quasi-symmetric functions. In the process, we introduce the notion of Schur-Weyl categories in order to describe rigidity theorems for cogroups in the category of associative algebras and related structures, as well as to account for the existence of natural operations (graded permutations) on them.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
