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Enriched categories are categories whose sets of morphisms are enriched withextra structure. Such categories play a prominent role in the study of highercategories, homotopy theory, and the semantics of programming languages. Inthis paper, we study univalent enriched categories. We prove that allessentially surjective and fully faithful functors between univalent enrichedcategories are equivalences, and we show that every enriched category admits aRezk completion. Finally, we use the Rezk completion for enriched categories toconstruct univalent enriched Kleisli categories.

Univalent Enriched Categories and the Enriched Rezk Completion