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.

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    $\begingroup$ I'm curious to see what the responses are, but I suspect this would be far from straightforward in general. The construction of the $Q_\ell$-homotopy type is outlined in the last section of Deligne's Weil II. $\endgroup$ – Donu Arapura Mar 29 '19 at 21:00
  • $\begingroup$ You may be interested in Toën's schematic homotopy types perso.math.univ-toulouse.fr/btoen/files/2015/02/AIM.pdf. You may also be interested in knowing that the étale homotopy type actually factors through the category of motivic spaces as in arxiv.org/pdf/1810.05544.pdf . You can't, however, stabilise that in the algebraic direction and get something reasonable. In general, I think that the same procedure would work for any other realisation as long as your $\infty$-topoi is locally connected (or maybe strongly connected) so that you have that shape functor. $\endgroup$ – user40276 Mar 29 '19 at 22:54
  • $\begingroup$ By the way, I don't know how Toën's theory of homotopy types fits into the usual motivic homotopy type theory. $\endgroup$ – user40276 Mar 29 '19 at 22:55

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