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

TL;DR

This paper develops a categorification framework for virtual braid groups and racks within symmetric categories, introducing self-distributive structures that produce braided objects and generalize algebraic identities.

Contribution

It provides a novel categorification of virtual braid groups and racks, and introduces self-distributive structures in symmetric categories, linking algebraic identities to braiding.

Findings

Categorification of virtual braid groups via locally braided objects.

Definition of self-distributive structures in symmetric categories.

Homology theory for categorical SDS generalizing existing differentials.

Abstract

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.