Algebraically cofibrant and fibrant objects revisited

TL;DR

This paper extends results on transferred model structures to weak model categories, showing that algebraically cofibrant and fibrant objects can be equipped with Quillen equivalent structures, connecting all combinatorial weak model categories to fibrant ones.

Contribution

It generalizes transferred model structures to weak model categories and establishes Quillen equivalences linking all combinatorial weak model categories to fibrant ones.

Findings

Transferred weak model structures exist on algebraically cofibrant and fibrant objects.

Under certain conditions, these structures are classical model structures.

Every combinatorial weak model category is Quillen equivalent to a fibrant one.

Abstract

We extend all known results about transferred model structures on algebraically cofibrant and fibrant objects by working with weak model categories. We show that for an accessible weak model category there are always Quillen equivalent transferred weak model structures on both the categories of algebraically cofibrant and algebraically fibrant objects. Under additional assumptions, these transferred weak model structures are shown to be left, right or Quillen model structures. By combining both constructions, we show that each combinatorial weak model category is connected, via a chain of Quillen equivalences, to a combinatorial Quillen model category in which all objects are fibrant.