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In this paper we show that the strict and lax pullbacks of a 2-categoricalopfibration along an arbitrary 2-functor are homotopy equivalent. We give twoapplications. First, we show that the strict fibers of an opfibration model thehomotopy fibers. This is a version of Quillen's Theorem B amenable toapplications. Second, we compute the $E^2$ page of a homology spectral sequenceassociated to an opfibration and apply this machinery to a 2-categoricalconstruction of $S^{-1}S$. We show that if $S$ is a symmetric monoidal2-groupoid with faithful translations then $S^{-1}S$ models the groupcompletion of $S$.

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