Definition
Finite abelian category
A linear abelian category equivalent to finite-dimensional modules over a finite-dimensional algebra.
A finite abelian category over is a -linear abelian category that is -linearly equivalent to the category of finite-dimensional left modules over some finite-dimensional associative unital -algebra.
Intrinsic conditions
Equivalently, Hom spaces are finite-dimensional, every object has finite length, there are finitely many simple isomorphism classes, and there are enough projectives. Finite length means that an object has a finite filtration with simple successive quotients. Having enough projectives means that every object is the target of an epimorphism from a projective object; an object is projective when maps from it lift across every epimorphism.
“Finite” does not mean finitely many objects or finitely many arrows. Finite-dimensional vector spaces form a finite abelian category, but their dimensions are unbounded.
Semisimplicity is extra
Finite-dimensional modules over form a finite abelian category which is not semisimple. The regular module has the submodule , and its resulting short exact sequence does not split. In contrast, a Hom-finite semisimple -linear category with finitely many simple isomorphism classes is finite abelian.
References
P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (2015), Definitions 1.8.5–1.8.6; author-hosted text.