The dual of Brown representability for homotopy categories of complexes

TL;DR

This paper establishes that the Brown representability theorem for the dual of the homotopy category of complexes holds if and only if the underlying additive category has a product generator, linking categorical properties to representability.

Contribution

It characterizes when Brown's theorem applies to the dual homotopy category in terms of the existence of a product generator in the base additive category.

Findings

Brown representability holds for the dual homotopy category if and only if a product generator exists.

The paper connects categorical properties with representability in homotopy categories.

Provides a necessary and sufficient condition for the dual of the homotopy category to satisfy Brown's theorem.

Abstract

We call product generator of an additive category a fixed object satisfying the property that every other object is a direct factor of a product of copies of it. In this paper we start with an additive category with products and images, e.g. a module category, and we are concerned with the homotopy category of complexes with entries in that additive category. We prove that Brown representability theorem is valid for the dual of the homotopy category if and only if the initial additive category has a product generator.