Representable cohomology theories in motivic homotopy theory

I am reading Mazza's, Voevodsky's and Weibel's book Lecture Notes on Motivic Cohomology and have grown curious about the following question:

Which cohomology theories on $$Sm/k$$ are representable, i.e. appear as Hom-groups, in Voevodsky's construction of $$DM_{\text{Nis}}^{eff,-}(k, R)$$?

If we let $$X$$ be a scheme of finite type over $$k$$ and $$R$$ a commutative unital ring, then we have (by construction) motivic homology with coefficients in $$R$$ $$H_{n,i}(X, R) = \text{Hom}_{DM_{\text{Nis}}^{eff,-}}(R(i)[n], R_{\text{tr}}(X)),$$ and motivic cohomology $$H^{n,i}(X, R) = \text{Hom}_{DM_{\text{Nis}}^{eff,-}}(R_{\text{tr}}(X), R(i)[n]).$$ From this, we obtain the algebraic singular homology (and hence Suslin's singular homology) as $$H^{\text{sing}}_n(X,R) \cong H_{n,0}(X,R)$$ and, if $$X$$ is a smooth separated scheme of finite type over some perfect field $$k$$, the higher Chow groups $$CH^q(X, 2q-p) = H^{p,q}(X,\mathbb{Z}).$$

It is clear to me that such cohomology theories must satisfy certain properties:

• Mayer-Vietories
• $$\mathbb{A}^1$$-invarience
• Künneth-theorem
• ...

But do we have any other examples of cohomology theories that appear as Hom-groups in $$DM_{\text{Nis}}^{eff,-}(k, R)$$? How about

• Betti cohomology?
• $$l$$-adic cohomology?
• Crystalline cohomology?
• Algebraic de Rham cohomology?

Another way of framing the question is: How far off is Voevodsky's-Morel's $$DM_{\text{Nis}}^{eff,-}(k, R)$$ construction from being the category of pure/mixed motives in the sense of Grothendieck.

(I realize this might be a very hard question to which not much is known.)

• Another "obvious" requirement is that the cohomology theory must have Voevodsky transfers (i.e. factor through the category $\operatorname{Corr}_k$ of Voevodsky correspondences). Feb 27 '21 at 16:10

Recall that $$\mathrm{DM}(k)$$ can be described as the subcategory of $$\mathrm{SH}(k)$$ made of modules over the motivic cohomology. This implies that cohomologies which are representable in $$\mathrm{DM}(k)$$ receive a cycle class map, so they must admit additive Chern classes ($$c_1(L\otimes L')=c_1(L)+c_1(L')$$). In fact, I think this is an if and only if, in the sense that motivic cohomology is the universal among cohomologies with additive Chern classes, but I do not know right now if this is written anywhere in the motivic setting with integral coefficients. In any case, if you put $$R=\mathbb{Q}$$ then you can have any type of Chern classes, for example $$K$$-theory. This is all written at Cisinki-Déglise's Triangulated categories..., chapter 14, which is in general a great reference for many motivic results up until 2009.
Your question is more natural for $$\mathrm{SH}(k)$$, as it happens already in topology, and then you know your cohomology would belong to $$\mathrm{DM}(k)$$ depending on if it is a module over motivic cohomology or not. An answer to your question in the setting of $$\mathrm{SH}$$ is in the paper of Cisinski and Déglise, Mixed Weil cohomologies, at 2.1.5. Loosely speaking, for any "sheaf cohomology" they construct an object of $$\mathrm{SH}(k)$$ representing such cohomology. Their hypothesis must imply having additive Chern classes, since they are willing to define a mixed variant of Weil cohomologies (which have additive Chern classes), so their cohomologies are in $$\mathrm{DM}(k)$$. At the end of the paper you have their main examples.
• $DM$ is absolutely not a full subcategory of $SH$. It is, as you correctly mention, a category of modules. Feb 27 '21 at 10:37