Injective Spaces via Adjunction

Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relation x −→ x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from (enriched) Category Theory for the investigation of (for instance) topological spaces. In this paper we study cocompleteness, adjoint functors and Kan extensions in the context of topological theories. We show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on Set. This way we obtain enriched versions of known results about injective topological spaces and continuous lattices. Introduction The title of the present article is clearly reminiscent of the chapter Ordered sets via adjunctions by R. Wood [Woo04], where the theory of ordered sets is developed elegantly employing consequently the concept of adjunction. One of the fundamental aspects of our recent research is described by the slogan topological spaces are categories, and therefore can be studied using notions and techniques from (enriched) Category Theory. We hope to be able to show in this paper that concepts like module, colimit and adjointness can be a very useful tool for the study of topological spaces too. We should explain what is meant by “spaces are categories”. In his famous 1973 paper [Law73] F.W. Lawvere considers the points of a (generalised) metric space X as the objects of a category X and lets the distance d(x, y) ∈ [0,∞] play the role of the hom-set of x and y. In fact, the basic laws 0 ≥ d(x, x) and d(x, y) + d(y, z) ≥ d(x, z) remind us immediately to the operations “choosing the identity” and “composition” 1 −→ hom(x, x) and hom(x, y) × hom(y, z) −→ hom(x, z) of a category. Motivated by Lawvere’s approach, we consider the points of a topological space X as the objects of our category, and interprete the convergence x −→ x of an ultrafilter x on X to a point x ∈ X as a morphism in X. With this interpretation, the convergence relation (∗) −→: UX × X −→ 2 becomes the “hom-functor” of X. Clearly, we have to make here the concession that a morphism in X does not have just an object but rather an ultrafilter (of objects) as domain. This intuition is supported by 2000 Mathematics Subject Classification. 18A05, 18D15, 18D20, 18B35, 18C15, 54B30, 54A20.