Modules as exact functors

TL;DR

This paper presents a new perspective on modules as exact functors on small abelian categories, connecting algebraic and model-theoretic viewpoints, with illustrative examples.

Contribution

It introduces an alternative definition of modules as exact functors, linking algebraic and model-theoretic concepts, and demonstrates their equivalence to traditional definitions.

Findings

Modules can be characterized as exact functors on small abelian categories.

The new perspective aligns with the classical definition of modules.

Several examples illustrate the applicability of this approach.

Abstract

We can define a module to be an exact functor on a small abelian category. This is explained and shown to be equivalent to the usual definition but it does offer a different perspective, inspired by the notions from model theory of imaginary sort and interpretation. A number of examples are worked through.