  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 $\mathfrak{x}\to x$ 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.
