An example of a non-cofibrantly generated model category

TL;DR

This paper demonstrates that certain model categories, specifically those generated by a proper class of orbits, are not cofibrantly generated, including the category of maps between spaces.

Contribution

It provides a counterexample showing that some model categories cannot be cofibrantly generated, expanding understanding of model category structures.

Findings

Model category of diagrams of spaces generated by a proper class of orbits is not cofibrantly generated.

The category of maps between spaces can be given a non-cofibrantly generated model structure.

Abstract

We show that the model category of diagrams of spaces generated by a proper class of orbits is not cofibrantly generated. In particular the category of maps between spaces may be given a non-cofibrantly generated model structure.