  In fairly elementary terms this paper presents, and expands upon, a recent
result by Garner by which the notion of topologicity of a concrete functor is
subsumed under the concept of total cocompleteness of enriched category theory.
Motivated by some key results of the 1970s, the paper develops all needed
ingredients from the theory of quantaloids in order to place essential results
of categorical topology into the context of quantaloid-enriched category
theory, a field that previously drew its motivation and applications from other
domains, such as quantum logic and sheaf theory.
