Is there a ring stacky approach to $\ell$-adic or rigid cohomology?

Question

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

[Sim2] C. Simpson - The Hodge filtration on nonabelian cohomology

[Bh] B. Bhatt - Prismatic $F$-Gauges

[GR] D. Gaitsgory, N. Rozenblyum - Crystals and D-modules

[RG] R. Gregoric - The Crystalline Space and Divided Power Completion

[PS] M. Porta, F. Sala - Simpson’s shapes of schemes and stacks

Answer 1

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.