  We continue work of our earlier paper (Lewitzka and Brunner: Minimally
generated abstract logics, Logica Universalis 3(2), 2009), where abstract
logics and particularly intuitionistic abstract logics are studied. Abstract
logics can be topologized in a direct and natural way. This facilitates a
topological study of classes of concrete logics whenever they are given in
abstract form. Moreover, such a direct topological approach avoids the often
complex algebraic and lattice-theoretic machinery usually applied to represent
logics. Motivated by that point of view, we define in this paper the category
of intuitionistic abstract logics with stable logic maps as morphisms, and the
category of implicative spectral spaces with spectral maps as morphisms. We
show the equivalence of these categories and conclude that the larger
categories of distributive abstract logics and distributive sober spaces are
equivalent, too.
