Positively closed parametrized models

Abstract

We study positively closed and strongly positively closed $\mathcal{C}\to Sh(B)$ models, where $\mathcal{C}$ is a $(\kappa ,\kappa )$-coherent category and $B$ is a $(\kappa ,\kappa )$-coherent Boolean-algebra for some weakly compact $\kappa $. We prove that if $\mathcal{C}$ is coherent and $X$ is a Stone-space then positively closed, not strongly positively closed $\mathcal{C}\to Sh(X)$ models may exist.