  We establish two duality theorems which refine the classical Stone duality
between generalized Boolean algebras and locally compact Boolean spaces. In the
first theorem we prove that the category of left-handed skew Boolean algebras
whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to
the category of \'{e}tale spaces over locally compact Boolean spaces whose
morphisms are \'{e}tale space cohomomorphisms over continuous proper maps. In
the second theorem we prove that the category of left-handed skew Boolean
$\cap$-algebras whose morphisms are proper skew Boolean $\cap$-algebra
homomorphisms is equivalent to the category of \'{e}tale spaces with compact
clopen equalizers over locally compact Boolean spaces whose morphisms are
injective \'{e}tale space cohomomorphisms over continuous proper maps.
