Journal
ALGEBRAIC AND GEOMETRIC TOPOLOGY
Volume 10, Issue 4, Pages 1933-2008Publisher
GEOMETRY & TOPOLOGY PUBLICATIONS
DOI: 10.2140/agt.2010.10.1933
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Funding
- NSF [DMS-0501208]
- Spanish Ministerio de Educacion y Ciencia [SB2006-0085]
- Macquarie University [DP0558598]
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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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