Positive model theory of interpretations

TL;DR

This paper extends classical model theory results to the setting of coherent functors between categories, introducing new invariants and studying their properties in relation to positively closed models.

Contribution

It develops a positive model theory framework for coherent functors, including variants of the omitting types theorem and ultraproducts, with a new lattice-valued invariant.

Findings

Analogues of model theory results for coherent functors are established.

A new distributive lattice valued invariant characterizes positively closed models.

Functorial properties of the invariant are analyzed.

Abstract

We prove analogues of model theory results for $\mathcal{C}\to \mathcal{D}$ coherent functors, including variants of the omitting types theorem and some results on ultraproduct constructions. We introduce a distributive lattice valued invariant of $\mathcal{C}\to \mathbf{Set}$ coherent functors that vanishes precisely on positively closed models, then we study its functorial properties.