Representing geometric morphisms using power locale monads

TL;DR

This paper demonstrates that geometric morphisms between elementary toposes can be represented as specific adjunctions between locale categories, characterized by preservation of order enrichment and compatibility with power locale monads.

Contribution

It provides a new categorical characterization of geometric morphisms using adjunctions that preserve order and commute with power locale monads.

Findings

Geometric morphisms correspond to adjunctions preserving order enrichment.

Such adjunctions commute with double, upper, and lower power locale monads.

Characterization includes preservation of finite coproducts by right adjoints.

Abstract

It it shown that geometric morphisms between elementary toposes can be represented as adjunctions between the corresponding categories of locales. These adjunctions are characterised as those that preserve the order enrichment, commute with the double power locale monad and whose right adjoints preserve finite coproduct. They are also characterised as those adjunctions that preserve the order enrichment and commute with both the upper and the lower power locale monads.