Exactly Definable Categories

Abstract

Ž op . Given a skeletally small abelian category C , we denote by Ex C , Ab the category of exact additive functors from C op to the category Ab of abelian groups. We call a category exactly definable provided that it is Ž op . equivalent to Ex C , Ab for some skeletally small abelian category C. This definition and our interest in this type of categories are motivated Ž . by the following example. Let L be a ring, and denote by C L s Ž op . fp mod L , Ab the category of finitely presented functors from the category of finitely presented L-modules to the category of abelian groups. Ž . The C L is skeletally small abelian. In fact, it is the free abelian category w x over L viewed as a category with a single object 4 , and the functor