Consider the following fusion categorie $F(i)$ with integer parameter $i$. Simple objects are $1,a,A,B$ (where $a$ and $A$ are conjugates). Nontrivial fusion rules are $a\bigotimes{a}=A$ (and conjugate), $B\bigotimes{a}=B$ (and conjugate), $B\bigotimes{B}=1\bigoplus{a}\bigoplus{A}\bigoplus{i\*B}$ (i.e. multiplicity $i$). The system is well known for $i=2$ (all quantum dimensions are integer then - offhand I can't give the reference but <http://arxiv.org/abs/1401.7096> should be your friend) but the fusion rules work for any i.  

Naturally the question arises, when is $F(i)$ braided? (Actually even modular.) No such luck: for $i=\infty$. I have no problem at all with thinking $\infty$ as integer, but you are mathematicians :-)  

Question: Can you make any sense out of a fusion category with some infinity multiplicity(ies), maybe as limit to another object? (The twist values are well defined, BTW - 18th roots of unity.) As modular fusion categories are rare and this wacko won't show up in any Galois theory-based classification program, it might be of interest.



Fusion categories: If infinity were an integer
Such tensor categories are called "near-group categories": these are semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. In your case, the group is $\{1,a,A\}$, and is isomorphic to $\mathbb Z/3$.


The paper <http://arxiv.org/pdf/1401.1879v2.pdf> shows that there's only finitely many values of your parameter $i$ such that your fusion ring comes from a fusion category (excluding $i=\infty$, the maximum possible value is $i=6$). Thus, I would find it very surprising if there were a semisimple category (suitably generalised to allow for infinite direct sums of simple objects) that realises the case $i=\infty$.


