The Stacks project

Definition 60.9.1. In Situation 60.7.5.

1. The (small) crystalline site of $X$ over $(S, \mathcal{I}, \gamma )$, denoted $\text{Cris}(X/S, \mathcal{I}, \gamma )$ or simply $\text{Cris}(X/S)$ is the full subcategory of $\text{CRIS}(X/S)$ consisting of those $(U, T, \delta )$ in $\text{CRIS}(X/S)$ such that $U \to X$ is an open immersion. It comes endowed with the Zariski topology.

2. The topos of sheaves on $\text{Cris}(X/S)$ is denoted $(X/S)_{\text{cris}}$ or sometimes $(X/S, \mathcal{I}, \gamma )_{\text{cris}}$¹.