Sheaves on involutive quantales: Grothendieck quantales

We show that sheaves on Grothendieck quantales, which are the quantales that correspond to Grothendieck toposes, may be defined in several equivalent ways. The main topic of this paper is the characterization of internal posets and suplattices in the topos of sheaves on a Grothendieck quantale Q, in terms of categories enriched in a suitable quantaloid and by means of Q-modules. We thereby extend the existing results concerning locales.