Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Univalent Enriched Categories and the Enriched Rezk Completion

Niels van der Weide

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.

Univalent Enriched Categories and the Enriched Rezk Completion

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.
Paper Structure (10 sections, 10 theorems, 9 equations)

This paper contains 10 sections, 10 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. The Bicategory of Enriched Categories
  3. Examples of Enriched Categories
  4. General Examples
  5. Enrichments over Structures
  6. Smash Products
  7. Image Factorization
  8. The Rezk Completion
  9. The Rezk Completion via Yoneda
  10. The Rezk Completion via HITs

Key Result

Proposition 2.3

For a monoidal category $\mathsf{V}\xspace$, the type of categories together with a $\mathsf{V}\xspace$-enrichment is equivalent to the type of $\mathsf{V}\xspace$-enriched categories.

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: https://nmvdw.github.io/EnrichedCats /UniMath. CategoryTheory.EnrichedCats.Enriched.EnrichmentEquiv.html# enriched_precat_weq_cat_with_enrichment
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: https://nmvdw.github.io/EnrichedCats /UniMath. Bicategories.DisplayedBicats.Examples.EnrichedCats.html# make_is_invertible_2cell_enriched
  • proof
  • Theorem 2.8: https://nmvdw.github.io/EnrichedCats /UniMath. Bicategories.DisplayedBicats.Examples.EnrichedCats.html# is_univalent_2_bicat_of_enriched_cats
  • proof
  • ...and 30 more