An algebraic description of regular epimorphisms in topology

Recent work of Janelidze and Sobral on descent theory of finite topological spaces motivated our interest in ultrafilter descriptions of various classes of continuous maps. In earlier papers we presented such characterizations for triquotient maps and local homeomorphisms, here we do it for regular epimorphisms. To do so, we give an alternative description of the “obvious” reflection of pseudotopological spaces into topological spaces. Topological spaces, when presented as ultrafilter convergence structures, are examples of image-algebras introduced by Clementino and Tholen in “Metric, Topology and Multicategory—a Common Approach”. In this paper, we work in this general setting and hence obtain at once characterizations of regular epimorphisms between topological spaces, approach spaces and (generalized) metric spaces, as well as the characterization for preordered sets which motivated our work.