Definition 60.8.4. In Situation 60.7.5.

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

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

The

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

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