Purity and flatness in symmetric monoidal closed exact categories

Abstract

Let [Formula: see text] be a symmetric monoidal closed exact category. This category is a natural framework to define the notions of purity and flatness. When [Formula: see text] is endowed with an injective cogenerator with respect to the exact structure, we show that an object [Formula: see text] in [Formula: see text] is flat if and only if any conflation ending in [Formula: see text] is pure. Furthermore, we prove a generalization of the Lambek Theorem (J. Lambek, A module is flat if and only if its character module is injective, Canad. Math. Bull. 7 (1964) 237–243) in [Formula: see text]. In the case [Formula: see text] is a quasi-abelian category, we prove that [Formula: see text] has enough pure injective objects.