Berkovich $2$-motives and normed ring stacks

TL;DR

This paper establishes a precise categorical framework linking motivic homotopy theory, ring stacks, and analytic geometry, enabling new motivic realization functors and connecting Berkovich motives with ring stacks.

Contribution

It formulates and proves a categorical principle relating motivic kernels to ring stacks, including étale and analytic versions, advancing the understanding of motivic realizations.

Findings

The category of motivic kernels is generated by a homologically trivial smooth sutured ring stack.

Étale descent reduces to Kummer and Artin--Schreier conditions.

Analytic version connects Berkovich motives with ring stacks, enabling motivic realizations in analytic geometry.

Abstract

The de Rham stack construction of Simpson shows that D-modules are quasicoherent sheaves on a modified geometry. Drinfeld furthermore introduced the ring stack perspective (aka transmutation), which asserts that a coefficient theory is determined by a ring stack. Scholze proposed relating this idea to motivic realizations using $(\infty,2)$-categorical language. In this work, we formulate and prove a precise version of this principle: The presentable category of kernels of motivic homotopy theory is the linearly symmetric monoidal $(\infty,2)$-category that is freely generated by a homologically trivial smooth sutured ring stack. We also prove the \'etale version of this statement, reducing \'etale descent to the Kummer and Artin--Schreier conditions. Lastly, we prove an analytic version connecting Scholze's Berkovich motives and ring stacks with an absolute value. This is useful to construct motivic realization functors in analytic geometry, such as the Habiro and Hyodo--Kato realizations.