The definition reads that "A G-extension of a fusion category D is a G-graded fusion category C whose trivial component is equivalent to D." It seems like a priori there can be multiple G-extensions for the same fusion category D. Is that really the case (i.e. no reduction mechanism)? But there seems to be a "canonical one" at least, which has the same category D sitting on top of each component of the grading. Does this particular G-extension have a name?



Is the G-extension of a fusion category unique?
A complete reference for G-extension of fusion categories is <http://arxiv.org/abs/0909.3140> (see also <http://arxiv.org/abs/0911.0881>). The kind of examples that you are saying are $\mathcal D\boxtimes \text{Vec}\_G$, the Deligne product of $\mathcal D$ with Vec$\_G$.


