  We extend Thomason's homotopy colimit construction in the category of
permutative categories to categories of algebras over an arbitrary $\Cat$
operad and analyze its properties. We then use this homotopy colimit to prove
that the classifying space functor induces an equivalence between the category
of $n$-fold monoidal categories and the category of $\mathcal{C}_n$-spaces
after formally inverting certain classes of weak equivalences, where
$\mathcal{C}_n$ is the little $n$-cubes operad. As a consequence we obtain an
equivalence of the categories of $n$-fold monoidal categories and the category
of $n$-fold loop spaces and loop maps after localization with respect to some
other class of weak equivalences. We recover Thomason's corresponding result
about infinite loop spaces and obtain related results about braided monoidal
categories and 2-fold loop spaces.
