Symmetric Categories as a Means of Virtualization for Braid Groups and Racks

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 selfdistributivity, 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 via weakly simplicial objects, generalizing rack, bar and Leibniz differentials. Free virtual SDS and the faithfulness of corresponding representations of virtual braid groups/groupoids are also discussed.