The universal six-functor formalism

TL;DR

This paper establishes that Morel-Voevodsky's stable ^1-homotopy theory provides a universal framework for Grothendieck's six functor formalism, unifying various cohomological operations.

Contribution

It proves the ^1-homotopy theory offers the universal coefficient system underlying Grothendieck's six operations.

Findings

^1-homotopy theory is universal for cohomological functors

Grothendieck's six operations are realized within this framework

The formalism applies broadly in algebraic geometry and homotopy theory

Abstract

We prove that Morel-Voevodsky's stable $\mathbb{A}^1$-homotopy theory affords the universal coefficient system, giving rise to Grothendieck's six operations.