Some Properties of Internal Locale Morphisms Externalised

TL;DR

This paper investigates how internal locale morphisms in Grothendieck toposes can be understood externally, providing pointwise characterizations of certain morphisms and co-frame operations.

Contribution

It offers a novel external perspective on internal locale morphisms, characterizing surjective and embedding morphisms as well as co-frame operations in a pointwise manner.

Findings

Characterization of morphisms inducing surjective geometric morphisms

Characterization of geometric embeddings via external methods

Pointwise computation of co-frame operations on internal sublocales

Abstract

We study morphisms of internal locales of Grothendieck toposes externally: treating internal locales and their morphisms as sheaves and natural transformations. We characterise those morphisms of internal locales that induce surjective geometric morphisms and geometric embeddings, demonstrating that both can be computed `pointwise'. We also show that the co-frame operations on the co-frame of internal sublocales can also be computed `pointwise' too.