Categorical Künneth formulae

In various geometric situations, given two geometric objects X and Y and their product X x Y, invariants of X x Y turn out to be closely related to the ones of X and of Y. For example, this is true for the fundamental group of topological spaces. Similarly, the Künneth formula describes the cohomology of X x Y in terms of the one of X and the one of Y. One level up, one may ask for a categorical Künneth formula, which concerns the question, of relevance in the Langlands program, whether one can decompose the entire category of sheaves on X x Y into pieces coming from X and Y. In this talk, I will try and give an invitation to this topic and then report on past and ongoing joint work with Timo Richarz concerning this question for ell-adic Weil sheaves in characteristic p, a newly minted category of homological motives, and our current understanding of this question for motivic sheaves.