I'm reading "UNSTABLE MOTIVIC HOMOTOPY THEORY" by Kirsten Wickelgren and Ben Williams (<https://arxiv.org/pdf/1902.08857.pdf>). There they have a version of Whitehead's Theorem, namely Prop 2.3, which says that $f:\mathcal{X}\rightarrow \mathcal{Y}$ is a weak equivalence if and only if the morphisms on homotopy sheaves $\pi\_n^{\mathbb{A}^1}(\mathcal{X})\rightarrow \pi\_n^{\mathbb{A}^1}(\mathcal{Y})$ are bijections for all basepoints of $\mathcal{X}$.


Naively, one would expect that Whitehead's Theorem should show that weak equivalence and equivalence are the same for a "nice" subset of spaces. This seems to show that the name "weak equivalences" for schemes up to $\mathbb{A}^1$-homotopy is well-choosen. This confuses me. Is the notion of weak equivalence in $\mathbb{A}^1$-homotopy a stronger property then in the classical topological case? For instance, do we have
$$[T,\mathcal{X}]\_{\mathbb{A}^1}\cong [T,\mathcal{Y}]\_{\mathbb{A}^1}$$
if $f$ is a weak equivalence of schemes up to $\mathbb{A}^1$-homotopy and $T$ is some scheme?



Whitehead Theorem in $\mathbb{A}^1$-homotopy theory
The condition you've stated implies that the homotopy sheaves are equivalent, and it is implied by the map being a weak equivalence, so they are equivalent. You're nullifying $\mathbb{A}^1$ in the $\infty$-category of Nisnevich sheaves of spaces on the category of (affine) schemes (smooth over the base). This ultimately can be seen from the fact that $\mathbb{A}^1$-nullification is a left Bousfield localization.


