Take $X$ to be a smooth complex projective algebraic variety. The Riemann-Hilbert correspondence gives an equivalence of categories between the category of perverse sheaves on $X$ and the category of holonomic D-modules on $X$.
Now Lurie, in the lecture "Notes on Crystals and Algebraic D-Modules" (link), demonstrates that in the above setting the category of quasicoherent D-modules is equivalent to the category of crystals of quasicoherent sheaves, or equivalently quasicoherent sheaves on the crystalline site. So the obvious inclusion of the D-module categories will give us perverse sheaves as a subcategory of crystalline sheaves.
Can I in general think of perverse sheaves as a subcategory of sheaves on the crystalline site? None of the literature I have seen on perverse sheaves takes this point of view, so I am somewhat suspicious of it; is it written up anywhere?
