Notions of enriched purity

We introduce enriched notions of purity depending on the left class E of a factorization system on the base V of enrichment. Ordinary purity is given by the class of surjective mappings in the category of sets. Under specific assumptions, covering enrichment over quantale-valued metric spaces, ω-complete posets, and quasivarieties, we characterize the (λ, E)-injectivity classes of locally presentable V-categories in terms of closure under a class of limits, λ-filtered colimits, and (λ,E)-pure subobjects.