Boolean valued models, presheaves, and étalé spaces

Moreno Pierobon, Matteo Viale

Published 2020-06-26Version 1

Boolean valued models for a signature $\mathcal{L}$ are generalizations of $\mathcal{L}$-structures in which we allow the $\mathcal{L}$-relation symbols to be interpreted by boolean truth values; for example for elements $a,b\in\mathcal{M}$ with $\mathcal{M}$ a $\mathsf{B}$-valued $\mathcal{L}$-structure for some boolean algebra $\mathsf{B}$, $(a=b)$ may be neither true nor false, but get an intermediate truth value in $\mathsf{B}$. In this paper we expand and relate the work of Mansfield and others on the semantics of boolean valued models, and of Munro and others on the adjunctions between $\mathsf{B}$-valued models and $\mathsf{B}^+$-presheaves for a boolean algebra $\mathsf{B}$. In particular we give an exact topological characterization (the so called \emph{fullness property}) of which boolean valued models satisfy \L o\'s theorem (i.e. the general form of the forcing theorem which Cohen -- Scott, Solovay, Vopenka -- established for the special case given by the forcing method in set theory). We also give an exact categorial characterization of which presheaves correspond to \emph{full} boolean valued models in terms of the structure of global sections of their associated \'etal\'e space. To do so we introduce a slight variant of the sheafification process of a given presheaf by means of its embedding into an \'etal\'e space.