Unstable motivic and real-étale homotopy theory

Abstract

We prove that for any base scheme $S$, real étale 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 étale motivic localization is given by smashing with the telescope of the map $\rho :S^0\to {\mathbb{G}}_m$.

34 pages, comments welcome