Definition 60.8.4. In Situation 60.7.5.

1. A family of morphisms $\{ (U_ i, T_ i, \delta _ i) \to (U, T, \delta )\}$ of divided power thickenings of $X/S$ is a Zariski, étale, smooth, syntomic, or fppf covering if and only if

1. $U_ i = U \times _ T T_ i$ for all $i$ and

2. $\{ T_ i \to T\}$ is a Zariski, étale, smooth, syntomic, or fppf covering.

2. The big crystalline site of $X$ over $(S, \mathcal{I}, \gamma )$, is the category $\text{CRIS}(X/S)$ endowed with the Zariski topology.

3. 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.

[1] This clashes with our convention to denote the topos associated to a site $\mathcal{C}$ by $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

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