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

TL;DR

This paper develops an $ ext{A}^1$-homotopy theory framework for rigid analytic spaces, establishing a homotopy category with functorial properties, a six functor formalism, and applications to analytic K-theory and spectra.

Contribution

It introduces an $ ext{A}^1$-homotopy category for rigid analytic spaces, extending homotopy-theoretic tools to non-archimedean geometry with new functorial and formalism results.

Findings

Established an $ ext{A}^1$-invariant homotopy category for rigid analytic spaces.

Proved the existence of a six functor formalism in this setting.

Identified connective analytic K-theory with algebraic K-theory in the homotopy category.

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 identify connective analytic K-theory in the unstable homotopy category with both $\mathbb{Z}\times\mathrm{BGL}$ and the analytification of connective algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra.