Enriched concepts of regular logic

TL;DR

This paper develops a framework for enriched regular logic, extending universal algebra to enriched categories, and characterizes models of regular theories as enriched injectivity classes.

Contribution

It introduces enriched language and formulas, and provides a categorical characterization of models of regular theories in the enriched setting.

Findings

Defines enriched regular logic with conjunctions and existential quantifiers.

Characterizes models as enriched injectivity classes.

Relates the logic to enriched category theory through factorization systems.

Abstract

Building on our previous work on enriched universal algebra, we define a notion of enriched language consisting of function and relation symbols whose arities are objects of the base of enrichment. In this context, we construct atomic formulas and define the regular fragment of enriched logic by taking conjunctions and existential quantifications of those. We then characterize enriched categories of models of regular theories as enriched injectivity classes in the enriched category of structures. These notions rely on the choice of a factorization system on the base of enrichment which will be used to interpret relation symbols and existential quantifications.