Sheaves of structures, Heyting‐valued structures, and a generalization of Łoś's theorem
Hisashi Aratake · Mathematical logic quarterly · 2021
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 Łoś's theorem for Heyting‐valued structures. We also give a characterization of Heyting‐valued structures for which Łoś's theorem holds with respect to any maximal filter.