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Six-Functor Formalisms III: The construction and extension of 6FFs

Chirantan Chowdhury

TL;DR

The paper completes the program for abstract six-functor formalisms by establishing a robust toolkit: a partial adjoints framework via multisimplicial methods, an ∞-categorical compactification to realize exceptional pushforwards, and a DESCENT-style extension philosophy to propagate six-functor data from small to large geometric contexts. It defines and leverages the ∞-category of correspondences, both in standard and bisimplicial forms, to encode pullbacks, pushforwards, and tensorial operations coherently, and provides concrete conditions (Nagata and descent hypotheses) under which 3- and 6-functor formalisms extend along nice geometric and exceptional pairs. The results enable extending cohomological formalisms such as étale cohomology, D-modules, and motivic frameworks from schemes to stacks, diamonds, and related geometric objects, with applications in arithmetic geometry and motivic homotopy theory. Technically, the work synthesizes adjointability, compactification, and localization techniques to realize a flexible, model-independent pathway for constructing and extending enhanced operation maps in higher categories.

Abstract

This article is the last of the series of articles where we reprove the foundational ideas of abstract six-functor formalisms developed by Liu-Zheng. We prove the theorem of partial adjoints, which is a simplicial technique of encoding various functors altogether by taking adjoints along specific directions. Combined with the $\infty$-categorical compactification theorem from the previous article, we can construct abstract six-functor formalisms in reasonable geometric setups of our interest. We also reprove the simplified versions of the DESCENT program due to Liu-Zheng, which allows us to extend such formalisms from smaller to larger geometric setups.

Six-Functor Formalisms III: The construction and extension of 6FFs

TL;DR

The paper completes the program for abstract six-functor formalisms by establishing a robust toolkit: a partial adjoints framework via multisimplicial methods, an ∞-categorical compactification to realize exceptional pushforwards, and a DESCENT-style extension philosophy to propagate six-functor data from small to large geometric contexts. It defines and leverages the ∞-category of correspondences, both in standard and bisimplicial forms, to encode pullbacks, pushforwards, and tensorial operations coherently, and provides concrete conditions (Nagata and descent hypotheses) under which 3- and 6-functor formalisms extend along nice geometric and exceptional pairs. The results enable extending cohomological formalisms such as étale cohomology, D-modules, and motivic frameworks from schemes to stacks, diamonds, and related geometric objects, with applications in arithmetic geometry and motivic homotopy theory. Technically, the work synthesizes adjointability, compactification, and localization techniques to realize a flexible, model-independent pathway for constructing and extending enhanced operation maps in higher categories.

Abstract

This article is the last of the series of articles where we reprove the foundational ideas of abstract six-functor formalisms developed by Liu-Zheng. We prove the theorem of partial adjoints, which is a simplicial technique of encoding various functors altogether by taking adjoints along specific directions. Combined with the -categorical compactification theorem from the previous article, we can construct abstract six-functor formalisms in reasonable geometric setups of our interest. We also reprove the simplified versions of the DESCENT program due to Liu-Zheng, which allows us to extend such formalisms from smaller to larger geometric setups.
Paper Structure (17 sections, 17 theorems, 64 equations)

This paper contains 17 sections, 17 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Recollection of notations and results.
  3. Constructing functors using category of simplices..
  4. The $\infty$-categorical compactification.
  5. Partial adjoints.
  6. Adjointable squares.
  7. Statement and proof of partial adjoints.
  8. The $\infty$-category of correspondences.
  9. Definitions and properties.
  10. Correspondences and bisimplicial sets.
  11. Six-Functor Formalisms.
  12. Definitions
  13. Construction of abstract 6FF.
  14. Extending Six-Functor Formalisms.
  15. Extension along nice geometric pairs.
  16. ...and 2 more sections

Key Result

Theorem 1.0.3

Let $(\mathcal{C},\mathcal{E})$ be pair as above. Let $(\mathscr{I},\mathcal{P})$ be two subsets of edges with the following assumptions : Let with the following assumptions : Then $\mathcal{D}^{*\otimes}$ can be extended to an abstract $6$-functor formalism: such that $f \in I$, we have $f_!=f_{\#}$ and for $f \in \mathcal{P}$, we have $f_! =f_*$.

Theorems & Definitions (77)

  • Definition 1.0.1
  • Definition 1.0.2
  • Theorem 1.0.3: Simplified version of \ref{['mainconstructiontheorem']}
  • Definition 1.0.4
  • Theorem 1.0.5
  • Definition 2.1.1
  • Definition 2.1.3
  • Example 2.1.4
  • Remark 2.1.5
  • Definition 2.1.6
  • ...and 67 more