  We compute the $L^2$-Betti numbers of the free $C^*$-tensor categories, which
are the representation categories of the universal unitary quantum groups
$A_u(F)$. We show that the $L^2$-Betti numbers of the dual of a compact quantum
group $G$ are equal to the $L^2$-Betti numbers of the representation category
$Rep(G)$ and thus, in particular, invariant under monoidal equivalence. As an
application, we obtain several new computations of $L^2$-Betti numbers for
discrete quantum groups, including the quantum permutation groups and the free
wreath product groups. Finally, we obtain upper bounds for the first
$L^2$-Betti number in terms of a generating set of a $C^*$-tensor category.
