Topological Groupoid Quantales

We associate a canonical unital involutive quantale to a topological groupoid. When the groupoid is also etale, this association is compatible with but independent from the theory of localic etale groupoids and their quantales [9] of P. Resende. As a motivating example, we describe the connection between the quantale and the C*-algebra that both classify Penrose tilings, which was left as an open problem in [5].