  The basic notions of category theory, such as limit, adjunction, and
orthogonality, all involve assertions of the existence and uniqueness of
certain arrows. Weak notions arise when one drops the uniqueness requirement
and asks only for existence. The enriched versions of the usual notions involve
certain morphisms between hom-objects being invertible; here we introduce
enriched versions of the weak notions by asking that the morphisms between
hom-objects belong to a chosen class of "surjections". We study in particular
injectivity (weak orthogonality) in the enriched context, and illustrate how it
can be used to describe homotopy coherent structures.
