Klaus Mattis - p-Completions in unstable motivic homotopy theory

Young-HOM 22-02-2024
Klaus Mattis

Title: p-Completions in unstable motivic homotopy theory
Abstract: In homotopy theory, one uses the p-completion functors to simplify questions about spectra or anima to the much easier case of p-complete spectra or anima. A similar thing can be done in motivic homotopy theory: I will define the unstable p-completion functor of a general oo-topos and of the category of motivic spaces. Then I will sketch how one can translate classical results about unstable p-completion from homotopy theory to the motivic setting. In particular, I will explain how one can obtain for every nilpotent motivic space X a short exact sequence 0 \to L_0 \pi_n(X) \to \pi_n^p(X_p^\wedge) \to L_1 \pi_{n-1}(X) \to 0, which is analogous to the classical situation. Here, the L_i are versions of the derived p-completion functors, and \pi_n^p is a certain "p-completed homotopy sheaf".