🤖 AI Summary
This work addresses the Rezk completion problem for univalent enriched categories. Methodologically, it first introduces and studies *univalent enriched categories*—a novel categorical structure where univalence is internalized via enrichment. It then establishes an equivalence criterion: any essentially surjective and fully faithful functor between such categories is necessarily an equivalence. Building on this, the paper constructs a universal Rezk completion from an arbitrary enriched category into a univalent enriched category—yielding the first general completeness theorem for this class. Finally, it develops the *univalent enriched Kleisli category*, providing a new semantic framework for higher homotopical semantics and formal semantics of programming languages. The approach integrates homotopy type theory, enriched category theory, and models of higher categories, thereby filling a foundational gap in the theory of categorical completion within the univalent framework.
📝 Abstract
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.