Relative injectivity as cocompleteness for a class of distributors
Abstract: 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:math.CT/0804.0326] 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
Set. In this paper we generalise these results, studying cocompleteness
with respect to a given class of distributors. We show in par-
ticular that the description of several semantic domains presented in
[M. Escardó and B. Flagg, Semantic domains, injective spaces and
monads, Electronic Notes in Theoretical Computer Science 20 (1999)] can
be translated into the V-enriched setting.
