  In this work the notions of partial action of a weak Hopf algebra on a
coalgebra and partial action of a groupoid on a coalgebra will be introduced,
just as some important properties. An equivalence between these notions will be
presented. Finally, a dual relation between the structures of partial action on
a coalgebra and partial action on an algebra will be established, as well as a
globalization theorem for partial module coalgebras will be presented.
