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It is well known that if G is an \'etale topological groupoid then itstopology can be recovered as the sup-lattice generated by G-sets, i.e. by theimages of local bisections. This topology has a natural structure of unitalinvolutive quantale. We present the analogous construction for any non \'etalegroupoid with sober unit space G_0. We associate a canonical unital involutivequantale 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 theconstruction mentioned above by proving a representability theorem for thisclass of quantales, under a natural spatiality condition.

Groupoid Quantales: a non \'etale setting