Exactness of Bousfield localizations of simplicial presheaves and local lifting

TL;DR

This paper characterizes when weak equivalences in Bousfield localizations of simplicial presheaves can be described via a local lifting property, establishing a connection with the exactness of the localization.

Contribution

It provides a necessary and sufficient condition for the local lifting property characterization of weak equivalences in Bousfield localizations, linking it to the exactness of the localization.

Findings

Weak equivalences characterized by local lifting property in exact localizations

Exactness of localization is equivalent to local lifting property characterization

Provides criteria for when Bousfield localizations are exact

Abstract

We show that weak equivalences in a (cofibrantly generated) left Bousfield localization of the projective model category of simplicial presheaves can be characterized by a local lifting property if and only if the localization is exact.