Notions of enriched purity

Abstract

We introduce enriched notions of purity depending on the left class $\mathcal E$ of a factorization system on the base $\mathcal V$ 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, $\omega$-complete posets, and quasivarieties, we characterize the $(\lambda,\mathcal E)$-injectivity classes of locally presentable $\mathcal V$-categories in terms of closure under a class of limits, $\lambda$-filtered colimits, and $(\lambda,\mathcal E)$-pure subobjects.