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

Fiore, Thomas M¹ ; Paoli, Simona²

¹Department of Mathematics and Statistics, University of Michigan-Dearborn, 4901 Evergreen Road, Dearborn, MI 48128
²Department of Mathematics and Statistics, Penn State Altoona, 3000 Ivyside Park, Altoona, PA 16601-3760

Algebraic and Geometric Topology, Tome 10 (2010) no. 4, pp. 1933-2008

Résumé

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.

Détail

DOI : 10.2140/agt.2010.10.1933

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

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