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The "Galois Correspondence" for n-Stacks

Yuxiang Yao

TL;DR

The paper establishes a higher Galois-type correspondence for Deligne-Mumford $n$-stacks by first proving an exact equivalence between $n$-groupoids in finite étale covers of a connected scheme $X$ and equivariant $n$-groupoids in finite sets under the étale fundamental group $ ext{π}_1^{ ext{ét}}(X, ext{η})$. It then uses simplicial/hammock localization to derive an essentially surjective functor from $ ext{π}_1^{ ext{ét}}(X, ext{η}) ext{-}n ext{-Stacks}( ext{FinSets})$ to $ ext{DM-}n ext{-Stacks}/ ext{étale over }X$, clarifying that this is not a full $ ext{∞}$-categorical equivalence. The work further develops the framework of categories with covers and fibrant objects, and extends the correspondence to topological and smooth contexts, yielding analogous functors for finite covers in Cov$(X)$ and Diff$(X)$. The results illuminate explicit Galois-type correspondences for higher stacks and offer practical pathways for constructing moduli spaces (e.g., Hurwitz-type stacks) via higher-stack techniques.

Abstract

We prove an essentially surjective Galois-correspondence-like functor for $n$-stacks. More specifically, it gives an essentially surjective functor from the $\infty$-category of $n$-stacks of finite sets with an action of the fundamental group of $X$ to the $\infty$-category of Deligne-Mumford $n$-stacks finite étale over a connected scheme $X$.

The "Galois Correspondence" for n-Stacks

TL;DR

The paper establishes a higher Galois-type correspondence for Deligne-Mumford -stacks by first proving an exact equivalence between -groupoids in finite étale covers of a connected scheme and equivariant -groupoids in finite sets under the étale fundamental group . It then uses simplicial/hammock localization to derive an essentially surjective functor from to , clarifying that this is not a full -categorical equivalence. The work further develops the framework of categories with covers and fibrant objects, and extends the correspondence to topological and smooth contexts, yielding analogous functors for finite covers in Cov and Diff. The results illuminate explicit Galois-type correspondences for higher stacks and offer practical pathways for constructing moduli spaces (e.g., Hurwitz-type stacks) via higher-stack techniques.

Abstract

We prove an essentially surjective Galois-correspondence-like functor for -stacks. More specifically, it gives an essentially surjective functor from the -category of -stacks of finite sets with an action of the fundamental group of to the -category of Deligne-Mumford -stacks finite étale over a connected scheme .
Paper Structure (5 sections, 17 theorems, 25 equations)

This paper contains 5 sections, 17 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. The Category of $n$-Groupoids in a Category $\mathcal{C}$
  3. Categories of Fibrant Objects
  4. The Main Theorem and Its Corollary
  5. Topological and Smooth Analogues

Key Result

Theorem 1

Let $\eta$ be a geometric point in a connected scheme $X$. There is an exact equivalence of categories of fibrant objects Here $n\text{-}\mathrm{Grpds}(\mathrm{FEt}_X)$ is the $1$-category of $n$-groupoids in $\mathrm{FEt}_X$, the category of finite étale (surjective) covers over $X$; and $\pi_1^{\mathrm{\acute{e}t}}(X, \eta)\text{-}n\text{-}\mathrm{Grpds}(\mathrm{FinSets})$ is the category of $n

Theorems & Definitions (34)

  • Theorem 1: Galois Correspondence for $n$-groupoids
  • Corollary 2
  • Theorem 3
  • Definition 4: Kernel pairs and Effective Epimorphisms
  • Definition 5: Category with covers and finite limits
  • Remark 6
  • Definition 7: $n$-groupoids, Definition 2.5, Wolfson
  • Definition 8: $n$-Hypercovers, Definition 2.6, Wolfson
  • Definition 9: Category of Fibrant Objects
  • Definition 10: Exact Functors between CFOs
  • ...and 24 more