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

Fiore, Thomas M; Paoli, Simona

doi:10.2140/agt.2010.10.1933

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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

DOI: 10.2140/agt.2010.10.1933

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 $C a t$ . 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.

Keywords

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

References

Publication

Received: 31 August 2008

Revised: 1 April 2010

Accepted: 17 August 2010

Published: 29 September 2010

Authors

Thomas M Fiore
Department of Mathematics and Statistics University of Michigan-Dearborn 4901 Evergreen Road Dearborn, MI 48128
http://www-personal.umd.umich.edu/~tmfiore/
Simona Paoli
Department of Mathematics and Statistics Penn State Altoona 3000 Ivyside Park Altoona, PA 16601-3760
http://math.aa.psu.edu/~simona/

Volume 10, issue 4 (2010)