Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sheaves of categories and the notion of 1-affineness

Dennis Gaitsgory

TL;DR

The paper develops the notion of 1-affineness for prestacks by formalizing sheaves of categories and their relationship to quasi-coherent sheaves QCoh(Y). It introduces Γ^enh_Y and Loc_Y as adjoint functors linking ShvCat(Y) with QCoh(Y)-mod and uses Barr-Beck-Lurie monadicity criteria to determine when these functors are equivalences. The main contributions establish 1-affineness for many classes (e.g., qcqs algebraic spaces, finite-type algebraic stacks, and certain de Rham and classifying prestacks), while identifying natural counterexamples (e.g., ind-schemes and certain infinite-type classifying spaces). The results provide practical criteria to verify 1-affineness via descent, monadicity, and Čech techniques, enabling a broad understanding of when complex geometric objects are recoverable from QCoh data. This work advances a higher-categorical approach to geometric representation theory and local geometric Langlands, clarifying when categories of sheaves of categories can be reconstructed from monoidal QCoh data and how these structures behave under deformation, completion, and quotient constructions.

Abstract

We define the notion of 1-affineness for a prestack, and prove an array of results that establish 1-affineness of certain types of prestacks.

Sheaves of categories and the notion of 1-affineness

TL;DR

The paper develops the notion of 1-affineness for prestacks by formalizing sheaves of categories and their relationship to quasi-coherent sheaves QCoh(Y). It introduces Γ^enh_Y and Loc_Y as adjoint functors linking ShvCat(Y) with QCoh(Y)-mod and uses Barr-Beck-Lurie monadicity criteria to determine when these functors are equivalences. The main contributions establish 1-affineness for many classes (e.g., qcqs algebraic spaces, finite-type algebraic stacks, and certain de Rham and classifying prestacks), while identifying natural counterexamples (e.g., ind-schemes and certain infinite-type classifying spaces). The results provide practical criteria to verify 1-affineness via descent, monadicity, and Čech techniques, enabling a broad understanding of when complex geometric objects are recoverable from QCoh data. This work advances a higher-categorical approach to geometric representation theory and local geometric Langlands, clarifying when categories of sheaves of categories can be reconstructed from monoidal QCoh data and how these structures behave under deformation, completion, and quotient constructions.

Abstract

We define the notion of 1-affineness for a prestack, and prove an array of results that establish 1-affineness of certain types of prestacks.

Paper Structure

This paper contains 149 sections, 110 theorems, 600 equations.

Table of Contents

  1. The notion of sheaf of categories
  2. Quasi-coherent sheaves on a prestack
  3. Main results
  4. Methods
  5. Organization of the paper
  6. Conventions
  7. Acknowledgements
  8. Quasi-coherent sheaves of categories
  9. Definition of a quasi-coherent sheaf of categories
  10. Global sections
  11. Posing the problem
  12. Dualizability and compact generation
  13. Descent
  14. Statements of the results
  15. Algebraic spaces and schemes
  16. ...and 134 more sections

Key Result

Lemma 1.1.3

The functor $\operatorname{ShvCat}(-)$ takes colimits in $\operatorname{PreStk}$ to limits in $\infty\operatorname{-Cat}$.

Theorems & Definitions (163)

  • Remark 3.1
  • Lemma 1.1.3
  • Lemma 1.3.5
  • proof
  • Definition 1.3.7
  • Lemma 1.4.2
  • Corollary 1.4.3
  • Proposition 1.4.5
  • proof
  • Lemma 1.4.6: Lurie
  • ...and 153 more