Towards ${\mathbb{A}}^1$-homotopy theory of rigid analytic spaces

Abstract

To any rigid analytic space (in the sense of Fujiwara--Kato) we assign an ${\mathbb{A}}^1$-invariant rigid analytic homotopy category with coefficients in any presentable category. We show some functorial properties of this assignment as a functor on the category of rigid analytic spaces. Moreover, we show that there exists a full six-functor formalism for the precomposition with the analytification functor by evoking Ayoub's thesis. As an application, we prove mod $p$ rigidity for rigid spaces over ${\mathbb{Q}}_p$. Moreover, we prove the equivalence of ${\mathbb{A}}^1$-invariant sheaves and ${\mathbb{B}}^1$-invariant sheaves in the mod $\ell$ case.

31 pages. New section on mod $p$ rigidity for étale motives. Former section on representability of K-theory is moved into a new article