A stacky approach to crystalline (and prismatic) cohomology.

The stacky approach was originated by Bhatt and Lurie. (But the possible mistakes in my talk are mine.)

Let X be a scheme over F_p. Many years ago Grothendieck and Berthelot defined the notion of crystal on X; moreover, they defined the notion of crystalline cohomology of a crystal.

I will give several equivalent definitions of a stack X^{prism} such that a crystal on X is the same as a quasi-coherent O-module on X^{prism} and the crystalline cohomology of a crystal is just the cohomology of this O-module. The stack X^{prism} is algebraic (in a certain sense).

If time permits, I will explain how to modify the definition of X^{prism} to get prismatic cohomology (this is a new theory due to Bhatt-Scholze, in which X is an arbitrary p-adic formal scheme rather than an F_p-scheme).