additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
Let $k$ be a field, and let $\mathcal{C}$ be a $k$-linear abelian category (i.e. one whose Ab-enrichment is lifted to a Vect-enrichment). Then $\mathcal{C}$ is said to be finite (over $k$) if
For any two objects $a$, $b$ of $C$, the hom-object ($k$-vector space) $\hom(a, b)$ has finite dimension;
Each object $a$ is of finite length;
There are only finitely many simple objects in $C$, and each of them admits a projective presentation.
For any finite abelian category $C$, there exists a finite-dimensional $k$-algebra $A$ and an $k$-linear equivalence between $C$ and $A$-$Mod_{fd}$, the category of modules over $A$ that are finite-dimensional as vector spaces over $k$.
Pavel Etingof and Victor Ostrik, Finite Tensor Categories, Preprint 2003 (arXiv:math/0301027)
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, chapter 6 of Tensor categories, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (pdf
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