Quasicategorical Galois theories

The classification of 1-toposes given by [Joyal-Tierney '84] admits a proof with a `categorical Galois theorem' playing a central role, as discussed in Galois Theories by Francis Borceux.. Our main result, Theorem 2.4.3, refines this result to a `quasicategorical Galois theorem.' Before we state and prove our main result, we expand upon the Mathematical content, and applicability, of the 1-categorical analogue. Specifically, we show how to begin with a duality result, and produce an application of the 1-categorical Galois theorem which mirrors the classical Galois theory of fields, rings, and covering spaces. We also offer a broadly accessible motivation for the 1-categorical Galois theorem, and hence for our quasicategorical Galois theorem. Finally, the proof of our main result is written as explicitly as possible toward the eventual goal of a model-independent generalization.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Rennie, Robert Joseph

Contributors dc:contributor

• Rezk, Charles
• Stojanoska, Vesna
• Ando, Matthew
• Heller, Jeremiah

• quasicategorical Galois theorem
• higher category theory
• Galois theory
• duality
• quasicategory
• enhanced mapping space
• Segal groupoid
• Segal groupoid object
• internal presheaves
• splitting object
• splitting algebra

Statement dc:rights

• Copyright 2022 Robert Rennie

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/116237

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F116237