A Thomason model structure on the category of small $n$–fold categories

Abstract

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.