Infinitesimal and $B\_\infty$-algebras, finite spaces, and quasi-symmetric functions

Abstract

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.