An extension of Quillen’s Theorem B

Moerdijk, Ieke; Nuiten, Joost

doi:10.2140/agt.2020.20.1769

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An extension of Quillen's Theorem B

Ieke Moerdijk and Joost Nuiten

Algebraic & Geometric Topology 20 (2020) 1769–1794

DOI: 10.2140/agt.2020.20.1769

Abstract

We prove a general version of Quillen’s Theorem B, for actions of simplicial categories, in an arbitrary left Bousfield localization of the homotopy theory of simplicial presheaves over a site. As special cases, we recover a version of the group completion theorem in this general context, as well a version of Puppe’s theorem on the stability of homotopy colimits in an $\infty$ –topos, due to Rezk.

Keywords

Quillen's Theorem B, group completion, simplicial presheaf, Nisnevich site, Bousfield localization, Rezk descent

Mathematical Subject Classification 2010

Primary: 18F20, 55U35

Secondary: 18G30, 55U10

References

Publication

Received: 21 August 2018

Revised: 22 July 2019

Accepted: 24 September 2019

Published: 20 July 2020

Authors

Ieke Moerdijk
Mathematical Institute Utrecht University Utrecht Netherlands

Joost Nuiten
Institut Montpelliérain Alexander Grothendieck Université de Montpellier Montpellier France

Volume 20, issue 4 (2020)