Special Year Research Seminar

A category of motives is an axiomatic framework in which several cohomology theories from algebraic geometry are represented. While Morel and Voevodsky's classical framework of motivic homotopy theory focused on $A^1$-invariant cohomology theories, such as $l$-adic étale cohomology, the more recent developments in integral $p$-adic Hodge theory have motivated lots of progress towards a more general theory of non-$A^1$-motives in which $p$-adic cohomology theories, such as crystalline or prismatic cohomology, are also represented. In this talk, I want to explain how one can formulate a precise universality conjecture for motivic cohomology using the recent progress in prismatic cohomology and in non-$A^1$-invariant motives, and report on the known cases of this conjecture.

This is based on a joint work in progress with Denis-Charles Cisinski and Niklas Kipp.