  The aim of the present paper is to extend the dualizing object approach to
Stone duality to the non-commutative setting of skew Boolean algebras. This
continues the study of non-commutative generalizations of different forms of
Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson,
Lawson and Lenz, Resende, and also the current author. In particular, we
construct a series of dual adjunctions between the categories of Boolean spaces
and skew Boolean algebras, unital versions of which are induced by dualizing
objects ${0,1,..., n+1}$, $n\geq 0$. We describe Eilenberg-Moore categories of
the monads of our adjunctions and construct easily understood non-commutative
reflections of skew Boolean algebras, where the latter can be faithfully
embedded (if $n\geq 1$) in a canonical way. As an application, we answer the
question that arose in a recent paper by Leech and Spinks to describe the left
adjoint to their `twisted product' functor $\omega$.
