Unstable motivic and real-\'etale homotopy theory

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 $\rho: S^0 \to {\mathbb G}_m$.