nLab semitransitive coring

Let $k$ be a (commutative) field and $R$ a $k$-algebra.

An $R$-coring $C$ is semi-transitive if the category $M^C_f$ of $C$-comodules which are finitely generated as modules over $R$

• $M^C_f$ is artinian and noetherian (has all objects of finite composition length), all its objects are projective as $R$-modules, and all its hom-spaces are finite dimensional over $k$
• the category of all $C$-comodules is Ind-generated by $M^C_f$ (Every $C$-comodule is a filtered colimit of $C$-comodules finitely generated as $R$-modules)

It appears that semi-transitive corings are those corings which can be reconstructed from fiber functors on categories which are artinian, noetherian and with finite dimensional hom-spaces.

• A. Bruguières, Théorie tannakienne non commutative, Comm. Algebra 22, 5817–5860, 1994

• K. Szlachanyi, Fiber functors, monoidal sites and Tannaka duality for bialgebroids, arxiv/0907.1578