Berkovich $2$-motives and normed ring stacks

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.