In algebraic geometry, we have something called Weil cohomology theories, which formalize the notion of a "good" cohomology theory of smooth projective varieties. I believe that for $l$-adic cohomology, we have a functorial construction of $l$-adic homotopy type. In general, given an arbitrary Weil cohomology theory is there a more-or-less formal construction of a (pro-)homotopy type having the corresponding cohomology groups?

I believe that my question should have nothing to do with motives since I am happy to serve one cohomology theory at a time but maybe I am wrong.