Absolutely flat algebras in tensor categories

Abstract: We study the relation between semisimple algebras, artinian simple algebras, and artinian absolutely flat algebras in ind-completions of tensor categories (rigid abelian monoidal categories). Our main results show that the three types of algebras coincide, if we restrict to commutative algebras in certain (all, if we accept the main conjectures in the field) symmetric tensor categories of moderate growth. We also provide examples to show that in general these notions tend to diverge, contrary to the classical case of algebras over a field. In one appendix we give the first, to the best of our knowledge, example of a tensor category with finite-dimensional morphism spaces but objects of infinite length. In a second appendix we establish a generalisation of Deligne's theorem that tannakian categories are neutral over algebraically closed fields.