Topological representation of intuitionistic and distributive abstract logics

TL;DR

This paper establishes a topological framework for studying intuitionistic and distributive abstract logics, showing their categorical equivalences with certain spectral spaces, simplifying their representation.

Contribution

It introduces a topological approach to abstract logics, defining categories of intuitionistic and distributive logics and proving their equivalences with spectral space categories.

Findings

Categories of intuitionistic abstract logics and implicative spectral spaces are equivalent.

Distributive abstract logics are equivalent to distributive sober spaces.

Topological methods simplify the study of abstract logics.

Abstract

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.