An extension of Quillen’s Theorem B

We prove a general version of Quillen’s Theorem B, for actions of simplicial cate-gories, 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 1–topos, due to Rezk.