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I'd like to explain Bruce's answer a bit more. The fusion categories Bruce mentioned have non-integer Frobenius-Perron dimensions, so it is very easy to see that they are not categories of finite dime …

This is meant to explain a bit more some of the things that were already mentioned before. Quantization of Lie bialgebras is indeed a functor, as was shown in my work with Kazhdan. However, the Kons …

Here is a definition that I heard: a family of additive categories over a scheme $X$ (say, defined over a field $k$) is an $k$-linear additive category ${\mathcal C}$ equipped with a structure of a mo …

Maybe it is helpful to look at these lectures by Yekutieli: arXiv:0801.3233. There he discusses twisted deformations of algebraic varieties $X$ suggested by Kontsevich, i.e. deformations of the catego …

One of the simplest consequences of Sylow's first theorem is the Cauchy theorem, saying that a group of order divisible by a prime $p$ contains an element of order $p$. I'd like to point out that th …

It seems that Ben is nevertheless right that the answer to the first question is "NO". Let $G=SL(2,\Bbb C)$, and $B$ be the subgroup of lower triangular matrices. Then the inclusion $B\to G$ is an epi …