ENRICHED CONCEPTS OF REGULAR LOGIC

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 $\mathcal {V}$ . 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 $\mathcal {V}$ -categories of models of regular theories as enriched injectivity classes in the $\mathcal {V}$ -category of structures. These notions rely on the choice of an orthogonal factorization system $(\mathcal {E},\mathcal {M})$ on $\mathcal {V}$ which will be used, in particular, to interpret relation symbols and existential quantification.