We construct a cofibrantly generated Quillen model structure on the category of small –fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An –fold functor is a weak equivalence if and only if the diagonal of its –fold nerve is a weak equivalence of simplicial sets. This is an –fold analogue to Thomason’s Quillen model structure on . We introduce an –fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the –fold nerve. As a consequence, we completely prove that the unit and counit of the adjunction between simplicial sets and –fold categories are natural weak equivalences.
"A Thomason model structure on the category of small $n$–fold categories." Algebr. Geom. Topol. 10 (4) 1933 - 2008, 2010. https://doi.org/10.2140/agt.2010.10.1933