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

Abstract

In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers. This is a version of Quillen's Theorem B amenable to applications. Second, we compute the E 2 page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of S −1 S. We show that if S is a symmetric monoidal 2-groupoid with faithful translations then S −1 S models the group completion of S.