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

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.