Non-commutative Stone dualities

In the talk, I will outline several approaches to non-commutative generalizations of Stone duality that have been developed in recent years. The general idea can be very briefly explained as follows:   we consider an algebra that generalizes a Boolean algebra (or a distributive lattice, or a frame) and enquire how the dual topological (localic) object of the commutative structure can be upgraded to dualize the whole algebra. We present dualities for Boolean inverse semigroups, pseudogroups and their non-involutive analogues called complete restriction semigroups. These objects are dualized by some topological (or, more generally, localic) categories or groupoids. I am also going to explain the natural role of quantales in these dualities. The talk is based  on several papers authored by to Mark Lawson, Daniel Lenz, Pedro Resende and the speaker.