# Enriched Regular Theories

## TL;DR

This paper extends the classical theory of regular and exact categories to an enriched setting, establishing analogous foundational theorems and dualities that connect enriched categories with their models.

## Contribution

It introduces an enriched notion of regularity and exactness, and proves versions of key theorems relating categories and their models in this new context.

## Key findings

- Established an enriched version of Barr's embedding theorem.
- Provided an explicit characterization of the essential image in the enriched setting.
- Extended duality results between exact categories and their model categories to enriched categories.

## Abstract

Regular and exact categories were first introduced by Michael Barr in 1971; since then, the theory has developed and found many applications in algebra, geometry, and logic. In particular, a small regular category determines a certain theory, in the sense of logic, whose models are the regular functors into Set. Barr further showed that each small and regular category can be embedded in a particular category of presheaves; then in 1990 Makkai gave a simple explicit characterization of the essential image of the embedding, in the case where the original regular category is moreover exact. More recently Prest and Rajani, in the additive context, and Kuber and Rosick\'y, in the ordinary one, described a duality which connects an exact category with its (definable) category of models. Considering a suitable base for enrichment, we define an enriched notion of regularity and exactness, and prove a corresponding version of the theorems of Barr, of Makkai, and of Prest-Rajani/Kuber-Rosick\'y.