The Compactness Theorem for Enriched Logic

Abstract

The framework of enriched logic extends positive logic (also known as coherent logic) from sets to structures internal to a symmetric monoidal closed category $\mathcal{V}$. This framework was developed by Rosický and the second-named author. The main novelty of this logic is that the arities of the function and relation symbols are now objects in $\mathcal{V}$. We prove the compactness theorem for this enriched logic. That is, a set of $\mathcal{V}$-enriched sequents has a model (internal to $\mathcal{V}$) if every finite subset has a model. To achieve this, we define ultraproducts of enriched structures and prove enriched versions of Łos' theorem. We then develop some model-theoretic applications. We start with an enriched version of the model-theoretic method of diagrams, establishing a correspondence between homomorphisms whose domain is some enriched structure $A$ and models of the set $\operatorname{Diag}(A)$ of all enriched atomic formulas with parameters from $A$ that are true in $A$. Combining this with our compactness theorem, we prove amalgamation results, laying the foundations for enriched syntactic types (in the model-theoretic sense). We finish the paper with a collection of examples of $\mathcal{V}$ that our results apply to. Besides recovering the classical case when $\mathcal{V} = \mathbf{Set}$, these include many presheaf toposes, categories of monoid acts, the category of abelian groups and related additive categories, categories of (small) enriched categories, and the categories of posets and pseudometric spaces.