  This paper gives a new interpretation of the virtual braid group in terms of
a strict monoidal category SC that is freely generated by one object and three
morphisms, two of the morphisms corresponding to basic pure virtual braids and
one morphism corresponding to a transposition in the symmetric group. The key
to this approach is to take pure virtual braids as primary. The generators of
the pure virtual braid group are abstract solutions to the algebraic
Yang-Baxter equation. This point of view illuminates representations of the
virtual braid groups and pure virtual braid groups via solutions to the
algebraic Yang-Baxter equation. In this categorical framework, the virtual
braid group is a natural group associated with the structure of algebraic
braiding. We then point out how the category SC is related to categories
associated with quantum algebras and Hopf algebras and with quantum invariants
of virtual links.
