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Let $K$ be a field. Does the category of finitely generated $K$-modules have a nice characterization, for example as the unique abelian category satisfying a certain simple condition? For example, we know that:

- Every short exact sequence is split.
- The Euler characteristic of every bounded exact sequence is zero.

Are either of those enough to characterize the category?

fixed$K$, and that seems unlikely to me unless we explicitly code in $K$, e.g. by working with $K$-linear categories. $\endgroup$ – Qiaochu Yuan Oct 22 '13 at 23:59