Some contributions to presheaf model theory

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}.

💡 Research Summary

The paper “Some contributions to presheaf model theory” develops a systematic treatment of “pure” sheaf model theory, i.e., model theory where the underlying structures are sheaves (or presheaves) over a complete Heyting algebra Ω. The authors begin by recalling the basic order‑theoretic notions—partial orders, complete lattices, and complete Heyting algebras—providing explicit definitions and elementary lemmas (adjunction, modus ponens, distributivity) that will be used throughout.

Next they introduce presheaves over Ω as a set |M| equipped with a restriction operation a↾p (for p∈Ω) and an extent map E(a)∈Ω, satisfying natural monotonicity and compatibility conditions. Extensional (or separated) presheaves are singled out, and a canonical “extensionalization” process is mentioned, allowing the authors to assume all presheaves are extensional without loss of generality.

The core of Section 2 is the definition of L‑structures on presheaves: a first‑order language L is interpreted pointwise, with terms evaluated via the restriction maps and formulas forced to hold at a point a∈|M| with a truth value in Ω. The forcing relation r⁽ᴹ⁾(a,b) is defined by comparing the extents of a and b; Lemma 1.16–1.18 show that this relation respects the Heyting algebra operations and characterises extensionality. The authors also discuss adding unary connectives corresponding to elements of Ω, thereby enriching the language to talk directly about truth‑values.

Section 3 studies preservation of forced formulas under homomorphisms between L‑structures. Two notions of morphism are considered: L‑morphisms (preserving the interpretation of symbols) and L‑monomorphisms (injective L‑morphisms). The authors prove a series of preservation lemmas culminating in a strengthened version of Robinson’s diagram method (originally in