# Can one characterize the category of finite-dimensional vector spaces? [duplicate]

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:

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

Are either of those enough to characterize the category?

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What do you mean by the Euler characteristic of a complex of objects in an abelian category? Do you mean the alternating sum of the images in the Grothendieck group, and if so, isn't this always equal to zero? –  Qiaochu Yuan Oct 22 '13 at 19:20
I'm also not sure how you expect to recover the field $K$. Are we secretly talking about $K$-linear categories or are you happy to characterize these categories as $K$ runs over all fields? –  Qiaochu Yuan Oct 22 '13 at 19:28
See my question mathoverflow.net/questions/118246/…, which has a very satisfactory answer. –  David Roberts Oct 22 '13 at 23:45
@Fernando: that's not what I mean. The wording of the OP's question suggests that he wants simple conditions that pin down $\text{FinVect}_K$ for fixed $K$, and that seems unlikely to me unless we explicitly code in $K$, e.g. by working with $K$-linear categories. –  Qiaochu Yuan Oct 22 '13 at 23:59