Unstable $p$-completion in motivic homotopy theory

Abstract

We define unstable $p$-completion in general $\infty$-topoi and the unstable motivic homotopy category, and prove that the $p$-completion of a nilpotent sheaf or motivic space can be computed on its Postnikov tower. We then show that the ($p$-completed) homotopy groups of the $p$-completion of a nilpotent motivic space $X$ fit into short exact sequences $0 \to \mathbb L_0 \pi_n(X) \to \pi_n^p(X_p^\wedge) \to \mathbb L_1 \pi_{n-1}(X) \to 0$, where the $\mathbb L_i$ are (versions of) the derived $p$-completion functors, analogous to the classical situation.