CATEGORICAL ASPECTS OF VIRTUALITY AND SELF-DISTRIBUTIVITY

Abstract

This paper revolves around two main results. First, we propose a "hom-set" type categorification of virtual braid groups and positive virtual braid monoids in terms of "locally" braided objects in a symmetric category (SC). This "double braiding" approach provides a rich source of representations, and offers a natural categorical interpretation for virtual racks and for the twisted Burau representation. Second, we define self-distributive (SD) structures in an arbitrary SC. SD structures are shown to produce braided objects in a SC. As for examples, we interpret the associativity and the Jacobi identity in a SC as generalized self-distributivity, thus endowing associative and Leibniz algebras with a (pre-)braiding. A homology theory of categorical SD structures is developed using the "braided" techniques from [Lebed, Homologies of algebraic structures via braidings and quantum shuffles, to appear in J. Algebra], generalizing rack, bar, Leibniz and other familiar complexes.