A Thomason Model Structure on the Category of Small n-fold Categories

TL;DR

This paper develops a model structure on small n-fold categories, establishing a Quillen equivalence with simplicial sets, and introduces an n-fold Grothendieck construction as a homotopy inverse.

Contribution

It constructs a cofibrantly generated Quillen model structure on small n-fold categories and proves its equivalence to simplicial sets, extending Thomason's work.

Findings

Established a Quillen model structure on n-fold categories.

Proved the Quillen equivalence with simplicial sets.

Introduced an n-fold Grothendieck construction as a homotopy inverse.

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.