Definition of Hurewicz map relating $SH(k)$ with $DM_\_^{eff}(k)$

Question

In "Motivic Homotopy Theory" on page 153 it is stated that there exists a canonical Hurewicz map relating the motivic stable homotopy category with the category $\operatorname{DM}_\_^{eff}(k)$. Unfortunatly, I was not able to find a definition of such a map and every attempt to define one myself ended in vain.

Answer 1

There's a free-forgetful adjunction between $SH_s(k)$ and $DM^{eff}(k)$, where $SH_s(k)$ is the category of $S^1$-spectra (as opposed to $\mathbb{P}^1$-spectra). The right adjoint simply takes a sheaf of chain complexes with transfers in $DM^{eff}(k)$ to its underlying sheaf of spectra (i.e. view chain complexes as spectra, and forget transfers).

You can then upgrade this adjunction to an adjunction between $SH(k)$ (= $SH_s(k)$ with $\Sigma^\infty\mathbb{G}_m$ inverted) and $DM(k)$ (= $DM^{eff}(k)$ with $\mathbb{Z}(1)[1]$ inverted). This works because the left adjoint above is symmetric monoidal and sends $\Sigma^\infty\mathbb{G}_m$ to $\mathbb{Z}(1)[1]$. The motivic Hurewicz map is the unit of this adjunction.

To get a functor $DM^{eff}\to SH$ you would first map $DM^{eff}$ to $DM$ (this is an embedding if $k$ is perfect), and then use the right adjoint $DM\to SH$.

For detailed constructions with model categories see section 2.2 in Modules over motivic cohomology by Röndigs and Østvær.