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6-Functor Formalisms and Smooth Representations

Claudius Heyer, Lucas Mann

Abstract

The purpose of this article is threefold: Firstly, we propose some enhancements to the existing definition of 6-functor formalisms. Secondly, we systematically study the category of kernels, which is a certain 2-category attached to every 6-functor formalism. It provides powerful new insights into the internal structure of the 6-functor formalism and allows to abstractly define important finiteness conditions, recovering well-known examples from the literature. Finally, we apply our methods to the theory of smooth representations of $p$-adic Lie groups and, as an application, construct a canonical anti-involution on derived Hecke algebras generalizing results of Schneider--Sorensen. In an appendix we provide the necessary background on $\infty$-categories, higher algebra, enriched $\infty$-categories and $(\infty,2)$-categories. Among others we prove several new results on adjunctions in an $(\infty,2)$-category and in particular show that passing to the adjoint morphism is a functorial operation.

6-Functor Formalisms and Smooth Representations

Abstract

The purpose of this article is threefold: Firstly, we propose some enhancements to the existing definition of 6-functor formalisms. Secondly, we systematically study the category of kernels, which is a certain 2-category attached to every 6-functor formalism. It provides powerful new insights into the internal structure of the 6-functor formalism and allows to abstractly define important finiteness conditions, recovering well-known examples from the literature. Finally, we apply our methods to the theory of smooth representations of -adic Lie groups and, as an application, construct a canonical anti-involution on derived Hecke algebras generalizing results of Schneider--Sorensen. In an appendix we provide the necessary background on -categories, higher algebra, enriched -categories and -categories. Among others we prove several new results on adjunctions in an -category and in particular show that passing to the adjoint morphism is a functorial operation.

Paper Structure

This paper contains 54 sections, 216 theorems, 430 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Categorical background
  3. Definition and construction of 6-functor formalisms
  4. The category of kernels
  5. Applications to representation theory
  6. Notation and conventions
  7. Acknowledgements
  8. The category of correspondences
  9. Geometric setups
  10. Definition of the correspondence category
  11. The symmetric monoidal structure
  12. Self-enrichment
  13. A 2-categorical enhancement on geometric setups
  14. 6-functor formalisms
  15. 3-functor formalisms
  16. ...and 39 more sections

Key Result

Proposition 3

Let $\mathcal{V}$ be a cartesian monoidal category.

Figures (1)

  • Figure 1: The diagram depicts $\mathbbl{\Sigma}^n$ (all arrows) and $\mathbbl{\Lambda}^n$ (solid arrows) for $n=3$. The poset $\mathbbl{\Sigma}_2^3$ contains only the arrows pointing to the right.

Theorems & Definitions (674)

  • Example 1: Künneth formula
  • Example 2: Poincaré duality
  • Proposition 3
  • proof
  • Example 4
  • Theorem 5
  • proof
  • Remark 6
  • Definition 7
  • Definition 8
  • ...and 664 more