Unstable motivic and real-\'etale homotopy theory

Aravind Asok; Tom Bachmann; Elden Elmanto; Michael J. Hopkins

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arXiv · 2501.15651

Unstable motivic and real-\'etale homotopy theory

Aravind Asok·Tom Bachmann·Elden Elmanto·Michael J. Hopkins

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

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Aravind Asok, Tom Bachmann, Elden Elmanto, Michael J. Hopkins. 2025-01-26. Unstable motivic and real-\'etale homotopy theory. https://arxiv.org/abs/2501.15651

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