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

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 $\mathbb{Z}\times \mathrm{B}\mathrm{G}\mathrm{L}$ and the analytification of connective algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra.