Homotopy Equivalents of the strict and pullbacks of a 2-categoricalfibration along an arbitrary 2-functor

2-categorical opfibrations, Quillen's Theorem B, and $S^{-1}S$

We show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent.This is a version of quillen s theorem amenable to applications.We give two applications.First, we compute the page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of.Second, we show that if is a symmetric monoidal2-groupoid with faithful translations then models the groupcompletion of .