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.
| Original language | English |
|---|---|
| Pages (from-to) | 1933-2008 |
| Number of pages | 76 |
| Journal | Algebraic and Geometric Topology |
| Volume | 10 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 29 Sept 2010 |
Bibliographical note
Thomas M Fiore and Simona Paoli thank the Centre de RecercaMatematica in Bellaterra (Barcelona) for its generous hospitality, as it provided a fantastic working environment and numerous inspiring talks. The CRM Research
Program on Higher Categories and Homotopy Theory in 2007–2008 was a great
inspiration to us both.
We are indebted to Myles Tierney for the suggestion to use the weak equivalence
N.=X/
/X and the Weak Equivalence Extension Theorem (Theorem 9.30) of
Joyal–Tierney [52] in our proof that the unit and counit of (24) are weak equivalences.
We also thank Andre Joyal and Myles Tierney for explaining aspects of Chapter 6 of ´
their book [52] which were particularly helpful for Section 9 of this paper.
We thank Denis-Charles Cisinski for explaining to us his proof that the unit and counit
are weak equivalences in the Thomason structure on Cat, as this informed our Section 9.
We also thank Dorette Pronk for several conversations related to this project.
We also express our gratitude to an anonymous referee who made many excellent suggestions.
Thomas M Fiore was supported at the University of Chicago by NSF Grant DMS0501208. At the Universitat Autonoma de Barcelona he was supported by grant `
SB2006-0085 of the Spanish Ministerio de Educacion y Ciencia under the Programa ´
Nacional de ayudas para la movilidad de profesores de universidad e investigadores
espanoles y extranjeros. Simona Paoli was supported by Australian Postdoctoral ˜
Fellowship DP0558598 at Macquarie University. Both authors also thank the Fields Institute for its financial support, as this project began at the 2007 Thematic Program on Geometric Applications of Homotopy Theory at the Fields Institute.