Ever since Simpson's paper [Sim], it was observed that many different cohomology theories arise in the following way: we begin with our space $X$, we associate to it a stack $X\_\text{stk}$ (which depends on the chosen cohomology theory), and then the cohomology of $X$ coincides with the quasi-coherent cohomology of $X\_\text{stk}$.


We have even more! Very often cohomology theories come from a six-functor formalism $D(X)$ and we often have that $D(X)=D\_\text{qc}(X\_\text{stk})$.


As far as I know, this is known for: (Often there are many references discussing this, I put the first one that came to mind just to help the reader find something about it.)


* de Rham cohomology; [GR] (We can also recover the underlying Hodge filtration. [Bh §2.3])
* Crystalline cohomology; [RG]
* Prismatic cohomology; [Bh]
* Syntomic cohomology; [Bh]
* Dolbeault cohomology; [Sim2]
* Betti cohomology; [PS]
* Deligne cohomology (I think); [PS]


The most notable absences from this list seem to be étale / $\ell$-adic cohomology and rigid cohomology. **Is there a stacky approach to these cohomology theories as well?**


References:


[Sim] C. Simpson - [Homotopy over the complex numbers and generalized de Rham cohomology](https://www.tib.eu/en/search/id/BLCP%3ACN013382849/Homotopy-Over-the-Complex-Numbers-and-Generalized/)


[Sim2] C. Simpson - [The Hodge filtration on nonabelian cohomology](https://arxiv.org/pdf/alg-geom/9604005.pdf)


[Bh] B. Bhatt - [Prismatic $F$-Gauges](https://www.math.ias.edu/%7Ebhatt/teaching/mat549f22/lectures.pdf)


[GR] D. Gaitsgory, N. Rozenblyum - [Crystals and D-modules](https://arxiv.org/pdf/1111.2087.pdf)


[RG] R. Gregoric - [The Crystalline Space and Divided Power Completion](https://web.ma.utexas.edu/users/gregoric/CrystalMath.pdf)


[PS] M. Porta, F. Sala - [Simpson’s shapes of schemes and stacks](https://people.dm.unipi.it/sala/assets/pdf/porta_sala_shapes.pdf)



Is there a ring stacky approach to $\ell$-adic or rigid cohomology?
This is an interesting question.


First, I think the [PS] reference does not give the "correct" Betti stack. In my notes on 6 functors, I define a different stack $X\_B$ such that $D\_{\mathrm{qc}}(X\_B)$ is equivalent to (the left-completed version of) $D(X,\mathbb Z)$, for any locally compact Hausdorff space $X$. It represents the functor taking any scheme $S$ to the continuous functions from $|S|$ to $X$. The [PS] reference would only be able to see the locally constant sheaves on $X$ instead, and is effectively passing to the homotopy type instead.


Roughly speaking, a prerequisite for a stacky approach to some cohomology theory is that this cohomology theory satisfies a categorical Künneth formula: $D(X)\otimes\_{D(\ast)} D(Y)\cong D(X\times Y)$. At least, $X\mapsto X\_B,X\_{\mathrm{dR}}$ etc. all commute with finite limits (and I think this should always be true for such stacks), and the functor $X\mapsto D\_{\mathrm{qc}}(X)$ often takes fibre products to tensor products (of presentable stable $\infty$-categories). The latter may fail a little bit in general, but at least it should be "close" to being true. For example, for locally compact Hausdorff spaces it is true at least for finite-dimensional ones.


For $\ell$-adic sheaves, the categorical Künneth formula fails badly. Thus, one cannot really expect a stacky approach. However, one can (and I do) hope for a stack $X\_{\ell,n}$ such that $D\_{\mathrm{et}}(X,\mathbb Z/\ell^n)$ embeds fully faithfully into $D\_{\mathrm{qc}}(X\_{\ell,n})$.


In fact, if one assumes that $X$ lives over $\mathbb Q\_\ell$, and one is allowing stacks in almost schemes, then such a thing has been defined (implicitly) through the work of Lucas Mann on $p$(=$\ell$)-adic 6 functors.


So over $\mathbb Q\_\ell$, it exists, but I don't know how to descend it to $\mathbb Q$.


Edit: And for rigid cohomology, I explained a construction of such a stack in my course, I hope I will one day update the notes to include it.


