  We construct a cofibrantly generated Quillen model structure on the category
of small n-fold categories and prove that it is Quillen equivalent to the
standard model structure on the category of simplicial sets. An n-fold functor
is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak
equivalence of simplicial sets. This is an n-fold analogue to Thomason's
Quillen model structure on Cat. We introduce an n-fold Grothendieck
construction for multisimplicial sets, and prove that it is a homotopy inverse
to the n-fold nerve. As a consequence, we completely prove that the unit and
counit of the adjunction between simplicial sets and n-fold categories are
natural weak equivalences.
