ENRICHED CONCEPTS OF REGULAR LOGIC

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 V. In this context, we construct atomic formulas and define the regular fragment of our enriched logic by taking conjunctions and existential quantification of those. We then characterize V-categories of models
of regular theories as enriched injectivity classes in the V-category of structures.
These notions rely on the choice of an orthogonal factorization system (E,M) on
V which will be used, in particular, to interpret relation symbols and existential
quantification.