Cofibrant generation of pure monomorphisms

Abstract

We show that pure monomorphisms are cofibrantly generated—generated from a set of morphisms by pushouts, transfinite composition, and retracts—in any locally finitely presentable additive category. In particular, this is true in any category of R-modules. On the other hand, the classes of all monomorphisms and regular monomorphisms in a locally finitely presentable additive category need not be well-behaved.