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

Thomas M Fiore and Simona Paoli

Algebraic & Geometric Topology 10 (2010) 1933–2008

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.

higher category, $n$–fold category, Quillen model category, nerve, $n$–fold nerve, Grothendieck construction, $n$–fold Grothendieck construction, Thomason model structure, subdivision
Mathematical Subject Classification 2000
Primary: 18D05, 18G55
Secondary: 55U10, 55P99
Received: 31 August 2008
Revised: 1 April 2010
Accepted: 17 August 2010
Published: 29 September 2010
Thomas M Fiore
Department of Mathematics and Statistics
University of Michigan-Dearborn
4901 Evergreen Road
Dearborn, MI 48128
Simona Paoli
Department of Mathematics and Statistics
Penn State Altoona
3000 Ivyside Park
Altoona, PA 16601-3760