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：富足范畴是一类其态射集合被赋予额外结构的范畴。此类范畴在高阶范畴、同伦理论及编程语言语义学研究中扮演着重要角色。本文研究单态富足范畴。我们证明所有单态富足范畴间的本质满且全忠实函子均为等价，并证明每个富足范畴均存在Rezk完备化。最后，我们利用富足范畴的Rezk完备化构造单态富足Kleisli范畴。