Artin glueings of frames as semidirect products
arXiv:1907.05104 · doi:10.1016/j.jpaa.2020.106334
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.
12 pages. To be published in the Journal of Pure and Applied Algebra. Added more discussion about morphisms of split extensions