# When is an Abelian Category a Category of Vector Spaces?

Any small abelian category admits a fully faithful exact functor into $R$-${\bf mod}$ for some ring $R$. What requirements can be imposed on an abelian category so that it is equivalent to $\mathbb{F}$-${\bf vect}$ for some field $\mathbb{F}$? I'm looking for a sufficient set of conditions in terms of familiar requirements on abelian categories.

Note that such conditions would also have to apply to the category of modules over any ring Morita equivalent to a field. For example, $\mathbb{F}$-${\bf vect}$ is Morita equivalent to $Mat_{n \times n} (\mathbb{F})$-${\bf vect}$, and $Mat_{n \times n} (\mathbb{F})$ is not a field.

If one requires all objects be projective, then there is a fully faithful exact functor into $R$-${\bf mod}$ for a ring $R$ which is the product of matrix algebras over division rings. Could this help?

• The embedding theorem is not particularly relevant here, since you are not asking about a small category. – Eric Wofsey Apr 5 '17 at 18:57