Some contributions to presheaf model theory

TL;DR

This paper advances pure sheaf model theory by providing new preservation results, refining existing theorems, and demonstrating the accessibility of categories of presheaves and sheaves within an independence framework.

Contribution

It offers a comprehensive treatment of sheaf model theory, including interpretation, preservation, and category-theoretic properties, extending prior results and integrating into the AECats framework.

Findings

Strengthened preservation results for sheaf models.

Refined understanding of directed colimits in sheaf categories.

Categories of presheaves and sheaves are accessible and fit within the AECats framework.

Abstract

This paper makes contributions to ``pure'' sheaf model theory, the part of model theory in which the models are sheaves over a complete Heyting algebra. We start by outlining the theory in a way we hope is readable for the non-specialist. We then give a careful treatment of the interpretation of terms and formulae. This allows us to prove various preservation results, including strengthenings of the results of \cite{BM14}. We give refinements of Miraglia's work on directed colimits, \cite{M88}, and an analogue of Tarski's theorem on the preservation of $\forall_2$-sentences under unions of chains. We next show various categories whose objects are (pairs of) presheaves and sheaves with various notions of morphism are accessible in the category theoretic sense. Together these ingredients allow us ultimately to prove that these categories are encompassed in the AECats framework for independence relations developed by Kamsma in \cite{K22}.