The Unreasonable Efficacy of the Lifting Condition in Higher Categorical Galois Theory I: a Quasi-categorical Galois Theorem

Abstract

In (Borceux-Janelidze 2001) they prove a Categorical Galois Theorem for ordinary categories, and establish the main result of (Joyal-Tierney 1984), along with the classical Galois theory of Rings, as instances of this more general result. The main result of the present work refines this to a Quasicategorical Galois Theorem, by drawing heavily on the foundation laid in (Lurie 2024). More importantly, the argument used to prove the result is intended to highlight a deep connection between factorization systems (specifically the lex modalities of (Anel-Biedermann-Finster-Joyal 2021)), higher-categorical Galois Theorems, and Galois theories internal to higher toposes. This is the first part in a series of works, intended merely to motivate the lens and prove Theorem 3.4. In future work, we will delve into a generalization of the argument, and offer tools for producing applications.