An introduction to six-functor formalisms
Martin Gallauer
TL;DR
Six-functor formalisms organize how cohomology behaves across geometry by elevating cohomology to a categorical invariant equipped with six functors. The paper advocates a coefficient-system framework as a practical surrogate for a full six-functor formalism, proving that such systems underlie six-functor formalisms and can be extended from small to presentable via Ind-completion. It identifies the initial universal system as stable motivic homotopy theory $\mathsf{SH}$, and explains how various realizations (Betti, Hodge, Beilinson) arise as motivic coefficient systems. The discussion also sketches a rigid-analytic analogue $\mathsf{RigSH}$, illustrating how exceptional functoriality can be obtained by transfer from algebraic geometry through Raynaud’s theory. Together these perspectives unify cohomology theories, descent properties, and functoriality across schemes and beyond, with implications for motivic homotopy theory and non-archimedean geometry.
Abstract
These are notes for a mini-course given at the summer school and conference "The Six-Functor Formalism and Motivic Homotopy Theory" in Milan 9/2021. They provide an introduction to the formalism of Grothendieck's six operations in algebraic geometry and end with an excursion to rigid-analytic motives. The notes do not correspond precisely to the lectures delivered but provide a more self-contained account for the benefit of the audience and others. No originality is claimed.