# Is the G-extension of a fusion category unique?

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?

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If the words you're using all mean what I think they mean, then no, extensions aren't unique: you can find nonequivalent $G$-extensions of $\text{Vect}$, for example, by twisting the associator by a class in $H^3(G, \mathbb{C}^{\ast})$. – Qiaochu Yuan Nov 19 '12 at 0:56
Thanks! This is great to know! Fits exactly. :-) – Lucy Zhang Dec 16 '12 at 11:48

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$_G$.