Lax algebra meets topology

We combine two research directions of the past decade, namely the development of a lax-algebraic framework for categories of interest to topologists and analysts, and the exploration of key topological concepts, like separation and compactness, in an abstract category which comes equipped with an axiomatic notion of “closed” or “proper” map. Hence, we present various candidates for such notions in the context of the category of ( T , V ) -categories, with a Set -monad T = ( T , e , m ) laxly extended to the category of sets and V -valued relations, for a quantale V . Suitable categories of ordered sets, metric spaces, topological spaces, closure spaces, and approach spaces all fit into this framework and allow for applications of the the general theory.