In the Tannaka-Krein duality theory, the Krein theorem describes the conditions under which a given category $\varPi$ is a category of finite-dimensional representations of some compact group $G$:

$\varPi$ must be a subcategory in the monoidal category of finite-dimensional vector spaces (with the usual tensor product $\otimes$),

$\varPi$ must have a unit, i.e. an object $I$ such that $I\otimes X\cong X$ for all objects $X$ in $\varPi$,

every object $X$ of $\varPi$ can be decomposed into a sum of minimal objects, and

for any minimal objects $X$ and $Y$ the space of homomorphisms $\operatorname{Hom}_{\varPi}(X,Y)$ is either one-dimensional or equal to zero.

I believe there must be analogous results for different other Tannaka duality theories, which are constructed now for many situations, including the very general one where $\varPi$ is the category of representations of a monoid $A$ in a closed symmetric monoidal category $V$.

I think, I am reading wrong books, but I only know the Krein theorem for compact groups: even the formulation for algebras $A$ over $\mathbb C$ is not familiar to me. Can anybody enlighten me on what is known in this field?

**EDIT.** I apologize, I have poorly formulated this question. Apparently, it was necessary to directly criticize the article in nLab, but I underestimated the importance of this source. After Qiaochu Yuan's answer, the problem, as I see it, is that in the list of examples in this article there are no references to specific statements in the literature. I am sure that this is easily fixable, and I would be grateful if someone would provide these links (either here, or in the article itself).