  We prove that the category of boolean inverse monoids is dually equivalent to
the category of boolean groupoids. This generalizes the classical Stone duality
between boolean algebras and boolean spaces. As an instance of this duality, we
show that the boolean inverse monoid associated with the Cuntz groupoid is the
strong orthogonal completion of the polycyclic (or Cuntz) monoid and so its
group of units is a Thompson group.
