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Six-Functor Formalisms I : Constructing functors using category of simplices

Chirantan Chowdhury

TL;DR

The paper addresses constructing functors from a simplicial set $K$ to a given $\infty$-category $\mathcal{C}$ by gluing data along the category of simplices. It develops a toolkit based on marked simplicial sets, the Cartesian model structure, and an injective model structure on functor categories, introducing the mapping functor $\operatorname{Map}[K,\mathcal{C}]$ and proving its fibrancy to enable controlled gluings. The main result provides concrete criteria (weak contractibility of fibers $\mathcal{N}(n,\sigma)$ and compatibility with a base map $f'$) ensuring the existence of a global functor $f:K\to\mathcal{C}$ extending $f'$, thus formalizing a simplicial-set level version of compactification/gluing arguments. This theorem underpins reproving the $\ ext{∞}$-categorical compactification and the Enhanced Operation Map in forthcoming work, and supplies a robust mechanism for DESCENT-type extensions across larger contexts in the six-functor formalism.

Abstract

This article is first in a series of papers where we reprove the statements in constructing the Enhanced Operation Map and the abstract six-functor formalism developed by Liu-Zheng. In this paper, we prove a theorem regarding constructing functors between simplicial sets using the category of simplices. We shall reprove the statement using the language of marked simplicial sets and studying injective model structure on functor categories. The theorem is a crucial tool and will be used repeatedly in reproving the $\infty$-categorical compactification and constructing the so called Enhanced Operation Map in the forthcoming articles.

Six-Functor Formalisms I : Constructing functors using category of simplices

TL;DR

The paper addresses constructing functors from a simplicial set to a given -category by gluing data along the category of simplices. It develops a toolkit based on marked simplicial sets, the Cartesian model structure, and an injective model structure on functor categories, introducing the mapping functor and proving its fibrancy to enable controlled gluings. The main result provides concrete criteria (weak contractibility of fibers and compatibility with a base map ) ensuring the existence of a global functor extending , thus formalizing a simplicial-set level version of compactification/gluing arguments. This theorem underpins reproving the -categorical compactification and the Enhanced Operation Map in forthcoming work, and supplies a robust mechanism for DESCENT-type extensions across larger contexts in the six-functor formalism.

Abstract

This article is first in a series of papers where we reprove the statements in constructing the Enhanced Operation Map and the abstract six-functor formalism developed by Liu-Zheng. In this paper, we prove a theorem regarding constructing functors between simplicial sets using the category of simplices. We shall reprove the statement using the language of marked simplicial sets and studying injective model structure on functor categories. The theorem is a crucial tool and will be used repeatedly in reproving the -categorical compactification and constructing the so called Enhanced Operation Map in the forthcoming articles.
Paper Structure (11 sections, 14 theorems, 38 equations)

This paper contains 11 sections, 14 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Marked Simplicial Sets and Cartesian Model Structure
  3. Marked Simplicial Sets
  4. Cartesian Model Structure
  5. Functors from the Category of simplices
  6. Properties and model structure on $(\operatorname{Set}_{\Delta})^{\mathcal{J}}$.
  7. The category of simplices.
  8. The mapping functor.
  9. The main theorem
  10. Remarks on absolute pullbacks.
  11. Remarks on anodynes.

Key Result

Theorem 1.0.2

chowdhury2024sixfunctorformalismsii Let $F: \delta^*_2N(\operatorname{Sch'})^{\operatorname{cart}}_{P,O} \to \mathcal{D}$ be a functor where $\mathcal{D}$ is an $\infty$-category. Then there exists a functor $F': N(\operatorname{Sch}') \to \mathcal{D}$ such that the diagram \begin{tikzcd}

Theorems & Definitions (47)

  • Definition 1.0.1
  • Theorem 1.0.2
  • Theorem 1.0.4
  • Definition 2.1.1
  • Remark 2.1.3
  • Definition 2.1.4
  • Remark 2.1.5
  • Definition 2.2.1
  • Remark 2.2.2
  • Definition 2.2.3
  • ...and 37 more