  We develop a theory of localization for braid group representations
associated with objects in braided fusion categories and, more generally, to
Yang-Baxter operators in monoidal categories. The essential problem is to
determine when a family of braid representations can be uniformly modelled upon
a tensor power of a fixed vector space in such a way that the braid group
generators act "locally". Although related to the notion of (quasi-)fiber
functors for fusion categories, remarkably, such localizations can exist for
representations associated with objects of non-integral dimension. We
conjecture that such localizations exist precisely when the object in question
has dimension the square-root of an integer and prove several key special cases
of the conjecture.
