# Contents

## Definition

The functor $\mathcal{O}:OLoc^\op \to Frm$ from the opposite category of overt locales to the category of frames is a regular hyperdoctrine. This means that it has an internal logic, where given an overt locale $L$, we can define the composition of open relations $E, F \in \mathcal{O}(L \times L)$ as

$F \circ E \coloneqq \{(x, z):L \times L \vert \exists y:L.(x, y) \in E \wedge (y, z) \in F\}$

A preuniform locale is a overt locale $L$ equipped with a filter $\mathcal{E}$ on the frame of opens $\mathcal{O}(L \times L)$ such that for each $E \in \mathcal{E}$,

• $(x, x) \in E$

• there exists an $E^o \in \mathcal{E}$ such that $(x, y) \in E$ entails and is entailed by $(y, x) \in E^o$

• there exists an $F \in \mathcal{E}$ such that $F \circ F \leq E$