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I am trying to understand relatively simple examples of calculations in $\mathbb{A}^1$-homotopy theory to build intuition.

For example, if $S$ is a scheme over a field $k$, what is the set $[\mathbb{G}_m, S]_{\mathbb{A}^1}$ of $\mathbb{A}^1$-homotopy classes of morphisms from $\mathbb{G}_m$ to $S$ over $k$? $\mathbb{G}_m$ is the multiplicative group.

How does this generalize to $\mathbb{G}_m^n$, $n \in \mathbb{N}$? Also dually, what are $[S, \mathbb{G}_m^n]_{\mathbb{A}^1}$?

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    $\begingroup$ This is not an elementary question. It is easier if you replace the multiplicative group by the projective line. For instance, there is a nice and rather explicit motivic version of the degree, as in this paper of Cazanave: numdam.org/item/ASENS_2012_4_45_4_511_0 $\endgroup$
    – D.-C. Cisinski
    Commented Mar 22, 2021 at 9:04
  • $\begingroup$ @Denis-CharlesCisinski, I too had in mind to think of [π”Ύπ‘š,𝑆] as a kind of loop space and pointed-maps version of $[π”Ύπ‘š,𝑆]_{\mathbb{A}^1}$ as a kind of fundamental group. Does that makes some sense? Using $\mathbb{G}_m^n$, or for that matter $\mathbb{P}^n$, as probe spaces to explore the topology of $S$ just as one does with spheres $S^n$ in classical homotopy theory seems like a fruitful idea. Would love your comments on it. $\endgroup$
    – Arna
    Commented Mar 26, 2021 at 19:48

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