Extension Theory for Braided-Enriched Fusion Categories

For a braided fusion category V, a V-fusion category is a fusion category C equipped with a braided monoidal functor F : V -> Z(C). Given a fixed V-fusion category (C, F) and a fixed G-graded extension C subset of D as an ordinary fusion category, we characterize the enrichments (F) over tilde : V -> Z(D) of D that are compatible with the enrichment of C. We show that G-crossed extensions of a braided fusion category C are G-extensions of the canonical enrichment of C over itself. As an application, we parameterize the set of G-crossed braidings on a fixed G-graded fusion category in terms of certain subcategories of its center, extending Nikshych's classification of the braidings on a fusion category.