Unstable motivic and real-\'etale homotopy theory

TL;DR

This paper establishes an equivalence between real étale motivic homotopy theory and semialgebraic topology over any base scheme, revealing a new connection between algebraic and topological frameworks in motivic homotopy theory.

Contribution

It demonstrates that real étale motivic homotopy theory coincides with semialgebraic topology and characterizes the localization process via smashing with a specific telescope.

Findings

Equivalence of real étale motivic and semialgebraic topologies over any base scheme.

Localization given by smashing with the telescope of a specific map.

Provides a new perspective linking algebraic and topological motivic theories.

Abstract

We prove that for any base scheme $S$, real \'etale motivic (unstable) homotopy theory over $S$ coincides with unstable semialgebraic topology over $S$ (that is, sheaves of spaces on the real spectrum of $S$). Moreover we show that for pointed connected motivic spaces over $S$, the real \'etale motivic localization is given by smashing with the telescope of the map $\rho: S^0 \to {\mathbb G}_m$.