Positively closed $Sh(B)$-valued models

TL;DR

This paper investigates the properties of positively closed models valued in toposes, distinguishing between global and local notions, and explores their existence and characterization in the context of Boolean algebras and infinite logic.

Contribution

It introduces the distinction between positively closed and strongly positively closed topos-valued models, proves their coincidence in Set-valued models, and constructs examples in Boolean algebra-based toposes.

Findings

Positively closed and strongly positively closed models coincide in Set-valued models.

Existence of positively closed but not strongly positively closed models in certain toposes.

An alternative local property characterizes positively closed models in these settings.

Abstract

We study positively closed and strongly positively closed topos-valued models of coherent theories. Positively closed is a global notion (it is defined in terms of all possible outgoing homomorphisms), while strongly positively closed is a local notion (it only concerns the definable sets inside the model). For $\mathbf{Set}$-valued models of coherent theories they coincide. We prove that if $\mathcal{E}=Sh(B,\tau _{coh})$ for a complete Boolean algebra, then positively closed but not strongly positively closed $\mathcal{E}$-valued models of coherent theories exist, yet, there is an alternative local property which characterizes positively closed $\mathcal{E}$-valued models. A large part of our discussion is given in the context of infinite quantifier geometric logic, dealing with the fragment $L^g_{\kappa \kappa }$ where $\kappa $ is weakly compact.