Enriched weakness

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.