  A useful general concept of bialgebroid seems to be resolving itself in
recent publications; we give a treatment in terms of modules and enriched
categories. We define the term "quantum category". The definition of antipode
for a bialgebroid is less resolved in the literature. Our suggestion is that
the kind of dualization occurring in Barr's star-autonomous categories is more
suitable than autonomy (= compactness = rigidity). This leads to our definition
of quantum groupoid intended as a "Hopf algebra with several objects".
