Sheaves of structures and generalized ultraproducts

Abstract

Classical ultraproducts are constructed from indexed sets of structures, i.e., from sheaves of structures on discrete spaces. We generalize the construction so that the initial datum can be an arbitrary sheaf of structures. Boolean ultrapowers are obtained in the special case where the initial sheaf is a constant sheaf. In Part I, we review the relevant information about sheaves on topological spaces. In Part II, we define notions of forcing and weak forcing in the stalks of any sheaf. From any sheaf of structures P on a space L we construct a sheafP 0 on the spectrum of filters of the pseudoBoolean algebra (pBa) of opens of/. The prime stalk is a Lo~-type theorem that characterizes the formulas weakly forced in any stalk o f P 0 at a pri~ne filter. In Part III, we construct a sheaf P* on the Stone space of the complete Boolean algebra of regular opens of/. The stalks of P* are called the ultrastalks of P, and they are t2~e generalized ultraproducts. With each tfltrafilter in the cBa (i.e., each point in the Stone space) there corresponds a maximal filter in the pBa, and the corresponding stalks of P* and p0 are isomorphic, thereby yielding two constructions of the generalized ultraproducts. The ultrastalk generalizes the ~o~ ultraproduct theorem by characterizing truth in the stalks of P* (or, equivalently, in the stalks o f P 0 at maximal filters). In Part IV, a few extensions and applications are outlined. The primary purpose of this paper i~ to familarize the working model