Constructing cogenerators in triangulated categories and Brown representability

Abstract

For a triangulated category with products we develop a method for constructing a nice set of cogenerators, allowing us to prove a formal criterion in order to satisfy Brown representability for covariant functors. We apply this criterion for showing that both homotopy category of projective modules and homotopy category of projective objects in a category of functors from a small category to a module category satisfy this kind of representability. In particular, homotopy category of projective complexes satisfies Brown representability for covariant functors.