Consider a variety $X$ over a field $k$ and let $\ell$ be a prime different from the characteristic of $k$. One has the derived category $D(X, Q_{\ell})$ of $\ell$-adic sheaves. There are very important abelian subcategories (of perverse sheaves) corresponding to $t$-structures given by various perversity conditions.

Question: What is the precise crystalline analogue of $D(X, Q_{\ell})$ and the subcategories of perverse sheaves? Here I am thinking of crystalline as being some sort of "$p$-adic sheaves" where $p$ is the characteristic of $k$. Perhaps one needs to assume $k$ is perfect..

Thank you in advance for your help.