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Monoid Extensions, Relaxed Actions and Cohomology 10.17863/CAM.69869 Faul, Peter Johnstone, Peter In this thesis a particular class of monoid extensions are studied and characterized, the weakly Schreier split extensions. It is demonstrated that both Artin glueings of frames and λ-semidirect products of inverse semigroups are examples of these extensions. The characterization is given in terms of a generalization of an action. This makes the theory amenable to cohomological ideas. Specifically, a new class of extensions called cosetal extensions are introduced and characterized. When parameterized by this new notion of action, a Baer sim may be defined in these extensions giving rise to an analogue of the second cohomology group. Finally, a connection is made to the setting of toposes, exploiting the link between toposes and frames. University of Cambridge 2020-12-01 Thesis Doctoral Doctor of Philosophy (PhD) eng https://www.repository.cam.ac.uk/handle/1810/322412 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/3797e463-4692-4510-8f57-06007d36874f/download bd76c53f017483e691c2d8823371e5d3 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/91e07072-303c-447d-bf30-48f644299ac8/download 353adac0d1ebdfd65ab16480263c3c87 https://creativecommons.org/licenses/by/4.0/ Monoid Extension Schreier Topos Frame Artin glueing