Categorical foundations of topology with applications to quantaloid enriched topological spaces

Abstract

Based on ordered monads this paper uncovers the categorical basis of topology in terms of a categorical formulation of neighborhood axioms. Here dense subobjects, lower separation axioms and regularity receive a purely categorical representation. In the case of appropriate submonads of the double presheaf monad this theory is applied to quantaloid-enriched topological spaces which form a common framework for many valued topology as well as for non-commutative topology. As an illumination of this situation two examples are given: the first one is chosen from probability theory and has the following characteristics: Weak convergence of τ-smooth probability measures is topological. The Hausdorff separation axiom is valid. Dirac measures form a dense subset. The second example is related to operator theory and explains the topologization of spectra of non-commutative C*-algebras.