For several cohomology theories for schemes it is possible to construct a geometric model: for any suitable scheme $X$ it is a ring stack $\mathcal{H}(X)$, defined over the coefficient ring $R$ of the cohomology theory $H$ and functorial in $X$, whose coherent cohomology $R\Gamma(\mathcal{H}(X),\mathcal{O}) \cong H(X,R)$. In fact this is true even at the level of derived categories of sheaves: $D_{qc}(\mathcal{H}(X)) \cong D(X)$.
I learnt about this “geometrization” of cohomology theories from Peter Scholze’s lecture “Motives and ring stacks” where it is described for de Rham (due to Simpson), prismatic (due to Drinfeld and Bhatt-Lurie) and Betti (due to Scholze) theories.
The question is:
What are the geometric properties that such a cohomology stack must have? Is it expected that any mixed Weil cohomology theory with a categorical Künneth formula will have a geometric model of this sort? Is it possible to characterize these stacks in the category of (algebraic/analytic) stacks?
