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2010 A Thomason model structure on the category of small $n$–fold categories
Thomas M Fiore, Simona Paoli
Algebr. Geom. Topol. 10(4): 1933-2008 (2010). DOI: 10.2140/agt.2010.10.1933


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.


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Thomas M Fiore. Simona Paoli. "A Thomason model structure on the category of small $n$–fold categories." Algebr. Geom. Topol. 10 (4) 1933 - 2008, 2010.


Received: 31 August 2008; Revised: 1 April 2010; Accepted: 17 August 2010; Published: 2010
First available in Project Euclid: 19 December 2017

zbMATH: 1203.18014
MathSciNet: MR2728481
Digital Object Identifier: 10.2140/agt.2010.10.1933

Primary: 18D05 , 18G55
Secondary: 55P99 , 55U10

Keywords: $n$–fold category , $n$–fold Grothendieck construction , $n$–fold nerve , Grothendieck construction , higher category , nerve , Quillen model category , Subdivision , Thomason model structure

Rights: Copyright © 2010 Mathematical Sciences Publishers

Vol.10 • No. 4 • 2010
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