  To an inverse semigroup, we associate an \'etale groupoid such that its
actions on topological spaces are equivalent to actions of the inverse
semigroup. Both the object and the arrow space of this groupoid are
non-Hausdorff. We show that this construction provides an adjoint functor to
the functor that maps a groupoid to its inverse semigroup of bisections, where
we turn \'etale groupoids into a category using algebraic morphisms. We also
discuss how to recover a groupoid from this inverse semigroup.
