  Notions and techniques of enriched category theory can be used to study
topological structures, like metric spaces, topological spaces and approach
spaces, in the context of topological theories. Recently in [D. Hofmann,
Injective spaces via adjunction, arXiv:0804.0326 [math.CV]] the construction of
a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness
and to show monadicity of the category of injective spaces and left adjoints
over $\mathsf{Set}$. In this paper we generalise these results, studying
cocompleteness with respect to a given class of distributors. We show in
particular that the description of several semantic domains presented in [M.
Escard\'o and B. Flagg, Semantic domains, injective spaces and monads,
Electronic Notes in Theoretical Computer Science 20 (1999)] can be translated
into the $\mathsf{V}$-enriched setting.

Category Theory;General Topology