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Six-Functor Formalisms II : The $\infty$-categorical compactification

Chirantan Chowdhury

TL;DR

This work develops an ∞-categorical framework for the exceptional pushforward in the abstract six-functor formalism by encoding compactifications and cartesian squares as multisimplicial structures. It proves two extension theorems, A and B, corresponding to extensions along the commutative and cartesian directions, using a prior technical simplicial result and contractibility of ∞-categories of decompositions. The approach relies on constructing two combinatorial devices, the ∞-category of compactifications and the ∞-category of cartesianizations, and showing their weak contractibility to enable required liftings. Together, these results advance Liu–Zheng’s program by providing an ∞-categorical pathway to define and glue exceptional functors, laying groundwork for the full abstract six-functor formalism in later work.

Abstract

This paper is part of a series of articles in which we reproduce the statements regarding the abstract six-functor formalism developed by Liu-Zheng. In this paper, we prove a theorem, which is an $\infty$-categorical version for defining the exceptional pushforward functor in an abstract-six functor formalism. The article describes specific combinatorial simplicial sets related to compactifications and pullback squares. This theorem plays a key role in constructing the abstract six-functor formalism, which will be discussed in the forthcoming article.

Six-Functor Formalisms II : The $\infty$-categorical compactification

TL;DR

This work develops an ∞-categorical framework for the exceptional pushforward in the abstract six-functor formalism by encoding compactifications and cartesian squares as multisimplicial structures. It proves two extension theorems, A and B, corresponding to extensions along the commutative and cartesian directions, using a prior technical simplicial result and contractibility of ∞-categories of decompositions. The approach relies on constructing two combinatorial devices, the ∞-category of compactifications and the ∞-category of cartesianizations, and showing their weak contractibility to enable required liftings. Together, these results advance Liu–Zheng’s program by providing an ∞-categorical pathway to define and glue exceptional functors, laying groundwork for the full abstract six-functor formalism in later work.

Abstract

This paper is part of a series of articles in which we reproduce the statements regarding the abstract six-functor formalism developed by Liu-Zheng. In this paper, we prove a theorem, which is an -categorical version for defining the exceptional pushforward functor in an abstract-six functor formalism. The article describes specific combinatorial simplicial sets related to compactifications and pullback squares. This theorem plays a key role in constructing the abstract six-functor formalism, which will be discussed in the forthcoming article.

Paper Structure

This paper contains 17 sections, 20 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Multisimplicial, multi-marked and multi-tiled simplicial sets.
  3. Multisimplicial sets.
  4. Multi-marked and multi-tiled simplicial sets.
  5. Recollection of key results from chowdhury2023sixfunctorformalismsi.
  6. The global section functor.
  7. Category of simplices.
  8. The main theorem and road map for proving \ref{['thmA']} and \ref{['thmB']}.
  9. Extension along $p_{\operatorname{comm}}$
  10. The $\infty$-category of compactifications.
  11. Proof of Theorem A.
  12. Extension along $p_{\operatorname{cart}}.$
  13. The $\infty$-category of cartesianizations,
  14. Proof of Theorem B.
  15. Inner anodyne maps between partially ordered sets.
  16. ...and 2 more sections

Key Result

Theorem 1.0.3

Let $\mathcal{C}$ be an $\infty$-category and $\mathcal{E}_1,\mathcal{E}_2$ be a collection of edges in $\mathcal{C}$ with the following conditions: Then for any $\infty$-category $\mathcal{D}$, there exists a solution to the lifting problem: \begin{tikzcd} \dd^*_2\Ca^{\op{cart}}_{\E_1,\E_2} \arrow[d,"p"] \arrow[r,"g"] &\D\\ \Ca \arrow[ur,"g'",dotted] & {}. \end{tikzcd}

Theorems & Definitions (80)

  • Definition 1.0.1
  • Definition 1.0.2
  • Theorem 1.0.3
  • Theorem 1.0.4: Theorem A : Extension along $p_{\operatorname{comm}}$
  • Theorem 1.0.5: Theorem B : Extension along $p_{\operatorname{cart}}$
  • Definition 2.1.1
  • Remark 2.1.2
  • Example 2.1.5
  • Definition 2.2.1
  • Remark 2.2.2
  • ...and 70 more