Quasicategorical Galois theories

Rennie, Robert Joseph

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

Description

Title

Quasicategorical Galois theories

Author(s)

Rennie, Robert Joseph

Issue Date

2022-07-14

Director of Research (if dissertation) or Advisor (if thesis)

Rezk, Charles

Doctoral Committee Chair(s)

Stojanoska, Vesna

Committee Member(s)

• Ando, Matthew
• Heller, Jeremiah

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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.

Graduation Semester

2022-08

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Robert Rennie

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