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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

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A complete reference for G-extension of fusion categories is (see also The kind of examples that you are saying are $\mathcal D\boxtimes \text{Vec}_G$, the Deligne product of $\mathcal D$ with Vec$_G$.

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Hey, Cesar! Thanks. Sorry for being off for a while. So, we can also twist the Deligne product, I assume. Let me look a bit more carefully into the reference paper. But it's always good to be able to just discuss intuitive. Hope to see you again soon. –  Lucy Zhang Dec 16 '12 at 11:10

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