Towards $\mathbb{A}^1$homotopy theory of rigid analytic spaces
Abstract
To any rigid analytic space (in the sense of FujiwaraKato) 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 Ktheory in the unstable homotopy category with both $\mathbb{Z}\times\mathrm{BGL}$ and the analytification of connective algebraic Ktheory. As a consequence, we get a representability statement for coefficients in light condensed spectra.
 Publication:

arXiv eprints
 Pub Date:
 July 2024
 DOI:
 10.48550/arXiv.2407.09606
 arXiv:
 arXiv:2407.09606
 Bibcode:
 2024arXiv240709606D
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Algebraic Geometry;
 Mathematics  KTheory and Homology;
 14F42;
 19E99;
 14G22
 EPrint:
 47 pages