We compute the
-Betti
numbers of the free
-tensor
categories, which are the representation categories of the universal unitary quantum groups
. We show that
the
-Betti
numbers of the dual of a compact quantum group
are equal to the
-Betti numbers of the
representation category
and thus, in particular, invariant under monoidal equivalence.
As an application, we obtain several new computations of
-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
-Betti number in terms of a
generating set of a
-tensor
category.
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