Enriched categories are categories whose sets of morphisms are enriched with extra structure. Such categories play a prominent role in the study of higher categories, homotopy theory, and the semantics of programming languages. In this paper, we study univalent enriched categories. We prove that all essentially surjective and fully faithful functors between univalent enriched categories are equivalences, and we show that every enriched category admits a Rezk completion. Finally, we use the Rezk completion for enriched categories to construct univalent enriched Kleisli categories.

翻译：充实范畴是其态射集被赋予额外结构的范畴。这类范畴在高层范畴论、同伦论及编程语言语义学研究中扮演着重要角色。本文研究了单值充实范畴，证明了单值充实范畴间所有本质满射且全忠实函子均为等价，并揭示了每个充实范畴均存在Rezk完备化。最后，我们利用充实范畴的Rezk完备化构造了单值充实Kleisli范畴。