# Is it possible to classify all Weil cohomologies?

Weil cohomologies seem to be "natural" and useful cohomology theories. Wikipedia lists Betti, De Rham, l-adic etale, and crystalline cohomologies as examples of Weil cohomology. Do we have more of them? and is it plausible to classify all or some Weil cohomologies? Or more generally, classify these really good Grothendieck sites?

-

An interesting (yet curious) example is an ultraproduct of etale coomology with $\mathbb Z/\ell$-coefficients over all $\ell$ different from the characteristic. It was used by Gabber to show that $\ell$-adic cohomology is torision free for all but a fini for a finite number of $\ell$ (and for a fixed smooth and projective variety). – Torsten Ekedahl Jun 4 '10 at 12:50