Zoran Skoda Seminar Bologna, Torsors

Torsors for geometrically admissible actions of monoidal categories

(Bologna, April 8, 2024)

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 generalizations ? If a Hopf algebra H (generalizing the algebra of functions on a group) 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. (This lifted comonad is related to a distributive law; entwining structures appear in some related examples.) The comodules over this comonad are the analogues of H-equivariant sheaves and the 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.