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.)