Volume 10, issue 4 (2010)

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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
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.

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