2012/06/18 by Victoria Lebed, Lebed, Victoria · 1 citation
Mathematics · #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1206.3916
openalex publication_date 2012/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we propose, firstly, a categorification of virtual braid groups and groupoids in terms of "locally" braided objects in a symmetric category (SC), and, secondly, a definition of self-distributive structures (SDS) in an arbitrary SC. SDS are shown to produce braided objects in a SC. As for examples, we interpret associativity and the Jacobi identity in a SC as generalized self-distributivity, thus endowing associative and Leibniz algebras with a "local" braiding. Our "double braiding" approach provides a natural interpretation for virtual racks and twisted Burau representation. Homology of categorical SDS is defined, generalizing rack, bar and Leibniz differentials. Free virtual SDS and the faithfulness of corresponding representations of virtual braid groups/groupoids are also discussed.