  This paper extends the fundamental results of frame theory to a
non-commutative setting where the role of locales is taken over by \'etale
localic categories. This involves ideas from quantale theory and from semigroup
theory, specifically Ehresmann semigroups, restriction semigroups and inverse
semigroups. We establish a duality between the category of complete restriction
monoids and the category of \'etale localic categories. The relationship
between monoids and categories is mediated by a class of quantales called
restriction quantal frames. This result builds on the work of Pedro Resende on
the connection between pseudogroups and \'etale localic groupoids but in the
process we both generalize and simplify: for example, we do not require
involutions and, in addition, we render his result functorial. We also project
down to topological spaces and, as a result, extend the classical adjunction
between locales and topological spaces to an adjunction between \'etale localic
categories and \'etale topological categories. In fact, varying morphisms, we
obtain several adjunctions. Just as in the commutative case, we restrict these
adjunctions to spatial-sober and coherent-spectral equivalences. The classical
equivalence between coherent frames and distributive lattices is extended to an
equivalence between coherent complete restriction monoids and distributive
restriction semigroups. Consequently, we deduce several dualities between
distributive restriction semigroups and spectral \'etale topological
categories. We also specialize these dualities for the setting where the
topological categories are cancellative or are groupoids. Our approach thus
links, unifies and extends the approaches taken in the work by Lawson and Lenz
and by Resende.
