Overcategories and undercategories of model categories

TL;DR

This paper establishes model category structures on overcategories and undercategories of a given model category, providing proofs for properties like cofibrant generation, cellularity, and properness.

Contribution

It rigorously proves that overcategories and undercategories inherit key model category properties when the base category has them.

Findings

Overcategories and undercategories inherit model structures.

Proofs provided for cofibrant generation, cellularity, and properness.

Results extend the theory of model categories to related categorical constructs.

Abstract

If M is a model category and Z is an object of M, then there are model category structures on the category of objects of M over Z and the category of objects of M under Z under which a map is a cofibration, fibration, or weak equivalence if and only if its image in M under the forgetful functor is, respectively, a cofibration, fibration, or weak equivalence. It is asserted without proof in "Model categories and their localizations" that if M is cofibrantly generated, cellular, or proper, then so is the category of objects of M over Z. The purpose of this note is to fill in the proofs of those assertions and to state and prove the analogous results for undercategories.