Torsors for geometrically admissible actions of monoidal categories

In Tannakian formalism, groups and generalizations like groupoids and Hopf algebras can be reconstructed from the fiber functor which is a forgetful strict monoidal functor from its category of modules to the category of vector spaces. Can we do something similar for the actions of groups, their properties and the generalizations ? If a Hopf algebra H coacts on an algebra A by a Hopf action (that is, A becomes an H-comodule algebra) then the category of H-modules acts on the category of A-modules, this action strictly lifts the trivial action of the category of vector spaces on A-modules and also H lifts to a comonoid in H-modules inducing a comonad on the category of A-modules. The comodules over this comonad are the analogues of H-equivariant sheaves and a Galois condition can be stated in terms of affinity in the sense of Rosenberg. We propose taking these properties as defining for a general framework allowing for the definition of Galois condition/principal bundles/torsors in a number of geometric situations beyond the cases of Hopf algebras coacting on algebras. We also sketch how many other examples like coalgebra-Galois extensions and locally trivial nonaffine noncommutative torsors fit into this framework.