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In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.
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Is there a ring stacky approach to $\ell$-adic or rigid cohomology?
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 … 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 $p$(=$\ell$ …