Volume 20, issue 4 (2020)

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An extension of Quillen's Theorem B

Ieke Moerdijk and Joost Nuiten

Algebraic & Geometric Topology 20 (2020) 1769–1794
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