We construct a theory of (étale) Berkovich motives. This is closely related to Ayoub’s theory of rigid-analytic motives, but works uniformly in the archimedean and nonarchimedean setting. We aim for a self-contained treatment, not relying on previous work on algebraic or analytic motives. Applying the theory to discrete fields, one still recovers the étale version of Voevodsky’s theory. Two notable features of our setting which do not hold in other settings are that over any base, the cancellation theorem holds true, and under only minor assumptions on the base, the stable ∞ -category of motivic sheaves is rigid dualizable.
Peter Scholze (Thu,) studied this question.