The Stacks project

Remark 60.11.4. To formulate the general notion of a crystal we use the language of stacks and strongly cartesian morphisms, see Stacks, Definition 8.4.1 and Categories, Definition 4.33.1. In Situation 60.7.5 let $p : \mathcal{C} \to \text{Cris}(X/S)$ be a stack. A crystal in objects of $\mathcal{C}$ on $X$ relative to $S$ is a cartesian section $\sigma : \text{Cris}(X/S) \to \mathcal{C}$, i.e., a functor $\sigma $ such that $p \circ \sigma = \text{id}$ and such that $\sigma (f)$ is strongly cartesian for all morphisms $f$ of $\text{Cris}(X/S)$. Similarly for the big crystalline site.