  Braided deformations of (symmetric) monoidal categories are related to
Vassiliev theory by a direct generalization of well-known results relating
"quantum" knot invariants to Vassiliev invariants. The deformation theory of
braidings is subsumed by the deformation theory of monoidal functors, which
proves surprisingly rich: the deformation complex of a monoidal functor has the
same structure as the deformation complex of an algebra, including a pre-Lie
structure, from which it is see that the problem of deforming monoidal functors
(including braidings) is purely cohomological in nature.
