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In this paper we propose, firstly, a categorification of virtual braid groupsand groupoids in terms of "locally" braided objects in a symmetric category(SC), and, secondly, a definition of self-distributive structures (SDS) in anarbitrary SC. SDS are shown to produce braided objects in a SC. As forexamples, we interpret associativity and the Jacobi identity in a SC asgeneralized self-distributivity, thus endowing associative and Leibniz algebraswith a "local" braiding. Our "double braiding" approach provides a naturalinterpretation for virtual racks and twisted Burau representation. Homology ofcategorical SDS is defined, generalizing rack, bar and Leibniz differentials.Free virtual SDS and the faithfulness of corresponding representations ofvirtual braid groups/groupoids are also discussed.

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