Purity and flatness in symmetric monoidal closed exact categories

Abstract

Let A be a symmetric monoidal closed exact category. This category is a natural framework to define the notions of purity and flatness. We show that an object F in A is flat if and only if any conflation ending in F is pure. Furthermore, we prove a generalization of the Lambek Theorem ([La64]) in A. In the case A is a quasi-abelian category, we prove that A has enough pure injective objects.