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.
Keywords
higher category, $n$–fold category, Quillen model category,
nerve, $n$–fold nerve, Grothendieck construction, $n$–fold
Grothendieck construction, Thomason model structure,
subdivision
