# Janelidze's Categorical Galois Theory as a step in the Joyal and Tierney result

## TL;DR

This paper demonstrates how a simple case of Janelidze's categorical Galois theorem can be instrumental in proving the Joyal-Tierney theorem on representing Grothendieck toposes as localic groupoids, linking Galois theory and topos theory.

## Contribution

It shows that a trivial case of Janelidze's Galois theorem can be used as a key step in the proof of a major result by Joyal and Tierney, and provides a new proof of the general Galois theorem.

## Key findings

- A trivial case of Janelidze's Galois theorem aids in proving the Joyal-Tierney result.
- The trivial case can be extended to prove the general Galois theorem.
- A technical result about sliced adjunctions facilitates the proof.

## Abstract

We show that a trivial case of Janelidze's categorical Galois theorem can be used as a key step in the proof of Joyal and Tierney's result on the representation of Grothendieck toposes as localic groupoids. We also show that this trivial case can be used to prove the general categorical Galois theorem by using a rather pleasing technical result about sliced adjunctions.