# Infinitesimal and B∞-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.
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Type de document :
Article dans une revue
Journal of Pure and Applied Algebra, Elsevier, 2016, 220 (6), pp.2434-2458
Domaine :

https://hal.archives-ouvertes.fr/hal-00967351
Contributeur : Patras Frédéric <>
Soumis le : jeudi 3 septembre 2015 - 12:31:24
Dernière modification le : vendredi 12 janvier 2018 - 01:51:46
Document(s) archivé(s) le : vendredi 4 décembre 2015 - 11:12:46

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topologies_FINAL.pdf
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### Identifiants

• HAL Id : hal-00967351, version 2
• ARXIV : 1403.7488

### Citation

Loïc Foissy, Claudia Malvenuto, Frédéric Patras. Infinitesimal and B∞-algebras, finite spaces, and quasi-symmetric functions. Journal of Pure and Applied Algebra, Elsevier, 2016, 220 (6), pp.2434-2458. 〈hal-00967351v2〉

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