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Categorification of sheaf theory

Germán Stefanich

TL;DR

The paper develops a systematic framework to categorify presentable six-functor formalisms through higher-categorical correspondences, producing a sequence of categorified formalisms $n\\Sh$ that yield topological field theories valued in presentable higher categories. By enriching $n\\Corr(\\ccal)$ over $\\Corr(\\ccal)$ and constructing $n\\Sh^{\\sharp}$, it generalizes the factorization-homology viewpoint to general sheaf theories, addressing when $\\Sh(\\Maps(M,X))$ can be recovered as TFT data. Descent and affineness analyses provide practical tools to verify when these categorified theories behave like classical sheaf theories, with concrete results for $\\QCoh$, $\\IndCoh$, and Betti theories, and applications to structures such as Rozansky–Witten theory and geometric Langlands. The framework thus unifies six-functor formalisms, TFTs, and categorified sheaf theories, enabling new perspectives on microlocal and global aspects of sheaves of higher categories. Overall, it offers a formal pathway to extend factorization-homology computations to broad categorical settings and extract TFT-based invariants from generalized sheaf theories.

Abstract

We discuss a systematic procedure for categorifying presentable six-functor formalisms. Our main result produces, given the input of a representation of the $\infty$-category of correspondences of an $\infty$-category with finite limits $\mathcal{C}$, a compatible sequence of representations of the $(\infty,n)$-category of correspondences of $\mathcal{C}$ for every $n \geq 1$. As an application, we explain a general recipe for constructing topological field theories.

Categorification of sheaf theory

TL;DR

The paper develops a systematic framework to categorify presentable six-functor formalisms through higher-categorical correspondences, producing a sequence of categorified formalisms that yield topological field theories valued in presentable higher categories. By enriching over and constructing , it generalizes the factorization-homology viewpoint to general sheaf theories, addressing when can be recovered as TFT data. Descent and affineness analyses provide practical tools to verify when these categorified theories behave like classical sheaf theories, with concrete results for , , and Betti theories, and applications to structures such as Rozansky–Witten theory and geometric Langlands. The framework thus unifies six-functor formalisms, TFTs, and categorified sheaf theories, enabling new perspectives on microlocal and global aspects of sheaves of higher categories. Overall, it offers a formal pathway to extend factorization-homology computations to broad categorical settings and extract TFT-based invariants from generalized sheaf theories.

Abstract

We discuss a systematic procedure for categorifying presentable six-functor formalisms. Our main result produces, given the input of a representation of the -category of correspondences of an -category with finite limits , a compatible sequence of representations of the -category of correspondences of for every . As an application, we explain a general recipe for constructing topological field theories.

Paper Structure

This paper contains 6 sections, 14 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Conventions and notation
  4. Enrichment of nCorr(C)
  5. Construction of nSh(X)
  6. Descent and affineness

Key Result

theorem 1

For every algebraic stack $X$ over $k$ and every $n \geq 0$, there is a symmetric monoidal functor $\chi_{(n+1)\kr\QCoh, X}: n\kr\Cob \rightarrow (n+1)\kr\Pr_{\st, k}$ such that for every closed manifold $M$ of dimension $n$ we have an equivalence

Theorems & Definitions (28)

  • theorem 1
  • theorem 2
  • corollary 1
  • corollary 2
  • theorem 3
  • proposition 1
  • proof
  • corollary 3
  • proof
  • definition 1
  • ...and 18 more