Sheaves of Structures, Heyting-Valued Structures, and a Generalization of {\L}o\'s's Theorem

TL;DR

This paper explores sheaves of structures and Heyting-valued structures within categorical logic, establishing a generalized version of { extL}o's theorem and characterizing conditions for its universal validity.

Contribution

It systematically develops the theory of sheaves and Heyting-valued structures and extends { extL}o's theorem to this context, providing new insights into their logical behavior.

Findings

Proves a form of { extL}o's theorem for Heyting-valued structures.

Provides a characterization of structures where { extL}o's theorem holds universally.

Offers a categorical logic perspective on sheaves of structures.

Abstract

Sheaves of structures are useful to give constructions in universal algebra and model theory. We can describe their logical behavior in terms of Heyting-valued structures. In this paper, we first provide a systematic treatment of sheaves of structures and Heyting-valued structures from the viewpoint of categorical logic. We then prove a form of {\L}o\'s's theorem for Heyting-valued structures. We also give a characterization of Heyting-valued structures for which {\L}o\'s's theorem holds with respect to any maximal filter.