An extension of Quillen's Theorem B

TL;DR

This paper generalizes Quillen's Theorem B to actions of simplicial categories within localized homotopy theories, unifying several important theorems in homotopy theory and topos theory.

Contribution

It extends Quillen's Theorem B to a broader context involving simplicial categories and Bousfield localizations, connecting multiple foundational results.

Findings

Generalized Quillen's Theorem B for simplicial categories

Unified group completion and Puppe's theorems in a homotopical setting

Established new stability results for homotopy colimits in infinity-toposes

Abstract

We prove a general version of Quillen's Theorem B, for actions of simplicial categories, in an arbitrary left Bousfield localization of the homotopy theory of simplicial presheaves over a site. As special cases, we recover a version of the group completion theorem in this general context, as well a version of Puppe's theorem on the stability of homotopy colimits in an infinity-topos, due to Rezk.