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.
Keywords: enriched accessible categories, enriched purity, enriched injectivity classes
2020 MSC: 18D20, 18C35
Theory and Applications of Categories, Vol. 41, 2024, No. 58, pp 2058-2104.
Published 2024-12-22.
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