Note on a theorem of Bousfield and Friedlander

TL;DR

This paper revisits a classical localization theorem by Bousfield and Friedlander, removing the need for the model category to be right proper, by introducing a new lemma about morphism factorizations.

Contribution

It extends the theorem's applicability to non-right proper model categories through a novel lemma on arrow category morphism factorizations.

Findings

The theorem holds without the right properness assumption.

A new lemma on morphism factorizations in arrow categories is introduced.

The proof is simplified and generalized.

Abstract

We examine the proof of a classical localization theorem of Bousfield and Friedlander and we remove the assumption that the underlying model category be right proper. The key to the argument is a lemma about factoring in morphisms in the arrow category of a model category.