The Hurewicz map in motivic homotopy theory

For an $\A^1$-connected pointed simplicial sheaf $\sX$ over a perfect field $k$, we prove that the Hurewicz map $π_1^{\A^1}(\sX) \to H_1^{\A^1}(\sX)$ is surjective. We also observe that the Hurewicz map for $¶^1_k$ is the abelianisation map. In the course of proving this result, we also show that for any morphism $ϕ$ of strongly $\A^1$-invariant sheaves of groups, the image and kernel of $ϕ$ are also strongly $\A^1$-invariant.

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