A Categorical Model for the Virtual Singular Braid Monoid

A mathematical braid is a set of strands passing between two parallel planes in space, such that each strand passes through any parallel plane that lies between the two planes exactly once. The set of all braids with n strands has the algebraic structure of a group, called the braid group on n strands. This algebraic structure arises by defining multiplication for braids as stacking one braid on top of another and gluing together their endpoints. In this thesis, we expand this structure to include virtual singular braids. The resulting algebraic structure is that of a monoid, called the virtual singular braid monoid on n strands. We provide a new presentation of the resulting monoid using generators and relations. We also show that the virtual singular braid monoid can be described in terms of a tensor category, freely generated by four morphisms and one object. Three of these morphisms are abstract presentations of the generators of the virtual singular braid monoid. The fourth morphism corresponds to a transposition in the symmetric group. We prove that the set of morphisms of this category is isomorphic to the virtual singular braid monoid on n strands.