  It is well known that if G is an \'etale topological groupoid then its
topology can be recovered as the sup-lattice generated by G-sets, i.e. by the
images of local bisections. This topology has a natural structure of unital
involutive quantale. We present the analogous construction for any non \'etale
groupoid with sober unit space G_0. We associate a canonical unital involutive
quantale with any inverse semigroup of G-sets which is also a sheaf over G_0.
We introduce axiomatically the class of quantales so obtained, and revert the
construction mentioned above by proving a representability theorem for this
class of quantales, under a natural spatiality condition.

Quantum Algebra;General Topology;Rings and Algebras