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Applications of (higher) categorical trace I: the definition of AGCat

Dennis Gaitsgory, Nick Rozenblyum, Yakov Varshavsky

TL;DR

<3-5 sentence high-level summary> AGCat provides an algebro-geometric enhancement of DG categories by universalising a monoidal Shv formalism via the enhancement construction Enh and the pair (Corr(Sch), DGCat) with Shv(−). This framework fixes the non‑equivalence of the external tensor product in the constructible/ℓ‑adic setting and enables robust trace, duality, and functoriality theories for prestacks, leading to a tractable

Abstract

In this paper we record the formalism of algebro-geometric DG categories (in short AGCat) following a suggestion of V. Drinfeld. This formalism will be applied to ``real-world" problems in papers sequel to this one, [GRV2] and [GRV3].

Applications of (higher) categorical trace I: the definition of AGCat

TL;DR

<3-5 sentence high-level summary> AGCat provides an algebro-geometric enhancement of DG categories by universalising a monoidal Shv formalism via the enhancement construction Enh and the pair (Corr(Sch), DGCat) with Shv(−). This framework fixes the non‑equivalence of the external tensor product in the constructible/ℓ‑adic setting and enables robust trace, duality, and functoriality theories for prestacks, leading to a tractable

Abstract

In this paper we record the formalism of algebro-geometric DG categories (in short AGCat) following a suggestion of V. Drinfeld. This formalism will be applied to ``real-world" problems in papers sequel to this one, [GRV2] and [GRV3].
Paper Structure (69 sections, 68 theorems, 600 equations)

This paper contains 69 sections, 68 theorems, 600 equations.

Table of Contents

  1. What is this paper about?
  2. What is this good for?
  3. How is AGCat actually constructed?
  4. What else is done in this paper?
  5. Structure of the paper
  6. Conventions and notations
  7. Acknowledgements
  8. The enhancement construction
  9. Setup and construction
  10. Basic properties of the enhancement construction
  11. The universal property
  12. Change of source
  13. Construction of AGCat
  14. The category of correspondences
  15. The functor Shv(-)
  16. ...and 54 more sections

Key Result

Proposition 1.2.5

Suppose that every object of ${\mathbf{O}}$ is dualizable. Then the symmetric monoidal category $\operatorname{Enh}({\mathbf{O}},{\mathcal{V}})$ is closed; i.e. admits internal homs.

Theorems & Definitions (120)

  • Remark 2.5
  • Remark 3.4
  • Remark 3.6
  • Proposition 1.2.5
  • proof
  • Proposition 1.2.8
  • proof
  • Proposition 1.2.10
  • proof
  • Proposition 1.3.3
  • ...and 110 more