  The effectiveness of the aplication 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
$k^G$ on a $k$-linear category. 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 $k^G$.
