$B_\infty$-algebras, their enveloping algebras, and finite spaces
Résumé
Finite topological spaces are in bijective correspondence with quasi-orders 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 set of finite spaces carries naturally 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 result that we obtain is certainly that the linear span of finite spaces carries the structure of the enveloping algebra of a $B_\infty$--algebra.
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