Partial Hopf module categories

The effectiveness of the application of constructions in G-graded k-categories to the computation of the fundamental group of a finite dimensional k-algebra, alongside with open problems still left untouched by those methods and new problems arisen from the introduction of the concept of fundamental group of a k-linear category, motivated the investigation of H-module categories, i.e., actions of a Hopf algebra H on a k-linear category. The G-graded case corresponds then to actions of the Hopf algebra kG on a k-linear category, where kG is the dual group algebra of G. In this work we take a step further and introduce partial H-module categories. We extend several results of partial H-module algebras to this context, such as the globalization theorem, the construction of the partial smash product and the Morita equivalence of this category with the smash product over a globalization. We also present a detailed description of partial actions of kG.