MathOverflow Asked by curious math guy on September 29, 2020

In contrast to "classical" homotopy theory, in the motivic homotopy theory, we don’t have homotopy group but rather homotopy sheaves in the Nisnevich topology, which is associated to the presheaf

$$pi_n^{mathbb{A}^1}(mathcal{X}):Sm_krightarrow text{Ab},quad Umapsto [S^{i,j}wedge U_+,mathcal{X}]_{mathbb{A}^1}.$$

Morphisms of spaces $mathcal{X}rightarrow mathcal{Y}$ induce morphisms of homotopy sheaves $pi_n^{mathbb{A}^1}(mathcal{X})rightarrow pi_n^{mathbb{A}^1}(mathcal{Y})$. The Nisnevich topology has enough points,and thus an isomorphism of homotopy sheaves can be detected on stalks. So a natural question is what is the stalk of the homotopy sheaves? Is it enough to evalute $pi_n^{mathbb{A}^1}(mathcal{X})$ on the spectra of local henselian rings?

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