Purity and flatness in symmetric monoidal closed exact categories

TL;DR

This paper explores the concepts of purity and flatness within symmetric monoidal closed exact categories, establishing key characterizations and generalizations of classical theorems in this framework.

Contribution

It introduces a characterization of flat objects via pure conflations and generalizes the Lambek Theorem within symmetric monoidal closed exact categories.

Findings

Flat objects are characterized by pure conflations ending in them.

A generalization of the Lambek Theorem is proved in this setting.

In quasi-abelian categories, enough pure injective objects exist.

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.