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

TL;DR

This paper develops an $ abla$-categorical framework for the six-functor formalism, proving a key theorem for defining the exceptional pushforward functor using combinatorial simplicial sets related to compactifications.

Contribution

It introduces an $ abla$-categorical approach to the six-functor formalism and proves a foundational theorem for the exceptional pushforward functor.

Findings

Proves an $ abla$-categorical version of the exceptional pushforward theorem.

Defines combinatorial simplicial sets related to compactifications.

Establishes a foundation for the abstract six-functor formalism.

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.