Generalized Brown representability in homotopy categories

TL;DR

This paper proves that the homotopy category of a combinatorial stable model category is well generated and explores conditions under which objects are weak colimits of compact objects, linking to Brown representability.

Contribution

It establishes the well-generated nature of homotopy categories in combinatorial stable model categories and connects this to generalized Brown representability.

Findings

Homotopy category of combinatorial stable model categories is well generated.

Objects are iterated weak colimits of compact objects.

Relation between weak colimits and Brown representability.

Abstract

We show that the homotopy category of a combinatorial stable model category $\ck$ is well generated. It means that each object $K$ of $\Ho(\ck)$ is an iterated weak colimit of $\lambda$-compact objects for some cardinal $\lambda$. A natural question is whether each $K$ is a weak colimit of $\lambda$-compact objects. We show that this is related to (generalized) Brown representability of $\Ho(\mathcal K)$.