On categories of ordered sets with a closure operator
Josef Šlapal · Publicationes Mathematicae Debrecen · 2011
We define and study two categories of partially ordered sets endowed with a closure operator.The first category has order-preserving and continuous maps as morphisms and it is shown to be concretely isomorphic to a category of ordered sets endowed with a compatible preorder.The second category has closed maps as morphisms and it is proved to be cartesian closed.Consequences of these results for categories of closure spaces and, in particular, of topological spaces are discussed.