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 $S$.

42 pages

References in corpus (5)

• A Quillen's Theorem A for strict $\infty$-categories I: the simplicial proof
• The 2-dimensional stable homotopy hypothesis
• Stable Postnikov data of Picard 2-categories
• La théorie de l'homotopie des 2-catégories
• A Quillen's Theorem A for strict $\infty$-categories II: the $\infty$-categorical proof