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arXiv · 2303.11957

Notions of enriched purity

Jiří Rosický·Giacomo Tendas
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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.

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Jiří Rosický, Giacomo Tendas. 2023-03-21. Notions of enriched purity. https://arxiv.org/abs/2303.11957

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