Several interesting cohomology theories can be described through (analytic) ring stacks, e.g. de Rham, Hodge, crystalline, prismatic, Betti, and even etale cohomology under some restrictions on the base. In this talk, I will recall that to any 6-functor formalism one can associate a (presentable) symmetric monoidal (\infty,2)-category. Adopting an extreme Tannaka duality-point of view to formulate the result, I will observe that the symmetric monoidal (\infty,2)-category associated to the motivic 6-functor formalism classifies (certain) ring stacks. This picture helps to explain why one has to pass to analytic geometry to find such ring stacks. (For example, the algebraic de Rham stack of A¹ is not a ring stack of the required form, only the analytic de Rham stack is.)

Peter Scholze (MPIM Bonn)

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