# Artin glueings of frames as semidirect products

## TL;DR

This paper explores how Artin glueings of frames can be understood as semidirect products via weakly Schreier split extensions, revealing new algebraic structures and properties in the category of frames.

## Contribution

It characterizes Artin glueings as weakly Schreier split extensions and introduces an extension bifunctor, connecting geometric and algebraic aspects of frames.

## Key findings

- Artin glueings correspond to weakly Schreier split extensions in frames.
- Extension bifunctor is constructed from meet-semilattice homomorphisms.
- The paper discusses Baer sums and the failure of the split short five lemma.

## Abstract

Artin glueings provide a way to reconstruct a frame from a closed sublocale and its open complement. We show that Artin glueings can be described as the weakly Schreier split extensions in the category of frames with finite-meet preserving maps. These extensions correspond to meet-semilattice homomorphisms between frames, yielding an extension bifunctor. Finally, we discuss Baer sums, the induced order structure on extensions and the failure of the split short five lemma.