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

TL;DR

This paper explores 2-categorical opfibrations, demonstrating homotopy equivalences of pullbacks, providing a 2-categorical version of Quillen's Theorem B, and constructing the group completion of symmetric monoidal 2-groupoids via $S^{-1}S$.

Contribution

It introduces a homotopy-theoretic analysis of 2-categorical opfibrations, extending classical results and constructing the group completion in a 2-categorical context.

Findings

Strict and lax pullbacks are homotopy equivalent.

Strict fibers model homotopy fibers in opfibrations.

The $S^{-1}S$ construction models the group completion of symmetric monoidal 2-groupoids.

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 $S$.