On the construction of functorial factorizations for model categories

TL;DR

This paper develops categorical techniques to construct functorial factorizations in model categories that are not cofibrantly generated, with applications to enriched categories and correcting previous errors.

Contribution

It introduces algebraic methods for functorial factorizations in non-cofibrantly generated model categories, expanding the toolkit for constructing model structures.

Findings

Established Hurewicz-type model structures in enriched categories

Corrected previous errors in constructing model structures for spaces and spectra

Provided categorical frameworks applicable to various enriched categories

Abstract

We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use "algebraic" characterizations of fibrations to produce factorizations that have the desired lifting properties in a completely categorical fashion. We illustrate these methods in the case of categories enriched, tensored, and cotensored in spaces, proving the existence of Hurewicz-type model structures, thereby correcting an error in earlier attempts by others. Examples include the categories of (based) spaces, (based) G-spaces, and diagram spectra among others.