  We introduce the notion of a dilation for a partial representation (i.e. a
partial module) of a Hopf algebra, which in case the partial representation
origins from a partial action (i.e.a partial module algebra) coincides with the
enveloping action (or globalization). This construction leads to categorical
equivalences between the category of partial $H$-modules, a category of
(global) $H$-modules endowed with a projection satisfying a suitable
commutation relation and the category of modules over a (global) smash product
constructed upon $H$, from which we deduce the structure of a Hopfish algebra
on this smash product. These equivalences are used to study the interactions
between partial and global representation theory.
