 Open locale

An open, or overt, locale has a positivity map $\mathrm{Pos}:\mathcal{O}(X)\to \Omega$ left adjoint to the scalar inverse image ${!}^{\ast }:\Omega \to \mathcal{O}(X)$, satisfying Frobenius reciprocity. Thus $a\le {!}^{\ast }\mathrm{Pos}(a)$ and positivity preserves arbitrary joins. Classically every locale is overt, with positivity meaning $a\ne 0$.