Given a group G grading a k-linear category \mathcal{C}, a Galois covering of \mathcal{C} has been associated to
this data in [CM] using the "smash product" construction, which is provided of a G-action. Since the category of Galois coverings of \mathcal{C} is equivalent to the full subcategory whose objects are the coverings obtained in this way, the computation of the fundamental group of \mathcal{C} may be restricted to this subcategory. It is worth to notice that having a G-grading is the same as having a kG-coaction, or, whenever G is finite, a k^G-action. The theory developed in [CS] suggests that one should extend the notion of grading of a k-linear category to the notion of H-module category, where H is a Hopf algebra, as it has been successfully done for algebras. Another option is to consider not only actions of groups on \mathcal{C} but partial actions, generalizing in this way the theory of partial actions on k-algebras (see [AB] for more details and for bibliographical references). The guiding idea is that this may allow us to obtain more "coverings". Our aim in this article is to develope both points of view, considering thus what we call partial H-module linear categories, with a particular interest on k^G-module categories.

This talk is based on joint work with Edson R. Alvares, Marcelo M.S. Alves and Eliezer Batista.

References:
[AB] M.M.S. Alves and E. Batista; Enveloping Actions for Partial Hopf Actions, Comm. Algebra 38 (2010), 2872-2902.
[CM] C. Cibils and E. Marcos; Skew category, Galois covering and smash product of a category over a ring, Proc. Amer. Math. Soc. 134 (2006), no. 1, 39-50.
[CS] C. Cibils and A. Solotar; Galois coverings, Morita equivalence and smash extensions of categories over a field, Doc. Math. 11 (2006), 143-159.