2. How many fusion categories...

Problem 2.1.
[Davydov] Can we find all fusion categories with a given smallest simple object (which is not invertible)?
Remark. [Scott Morrison] For example, we know that the smallest possible fusion dimension $1/2(\sqrt{3}+\sqrt{7})$ is realized by the IzumiXuOstrik fusion category from [arXiv:1004.0665]. What other fusion categories contain an object with this dimension?


Problem 2.2.
[Wang] Is there an effective version of Ocneanu rigidity? Is there a subexponential bound on the number of unitary fusion categories with respect to $N$, the sum of all the fusion multiplicities $N_{i,j}^k$? 
Problem 2.3.
[Wenzl] How many fusion categories have the same given fusion rules? 
Problem 2.4.
What can you say about all fusion categories $\mathcal{C}$ for which $\#\{\dim(X)X\in\mathcal{C}\}=2?$
Cite this as: AimPL: Classifying fusion categories, available at http://aimpl.org/fusioncat.