The Stacks project

Remark 60.9.4 (Comparison with Zariski site). In Situation 60.7.5. The functor (60.9.1.1) is continuous, cocontinuous, and commutes with products and fibred products. Hence we obtain a morphism of topoi

\[ u_{X/S} : (X/S)_{\text{cris}} \longrightarrow \mathop{\mathit{Sh}}\nolimits (X_{Zar}) \]

relating the small crystalline topos of $X/S$ with the small Zariski topos of $X$. See Sites, Section 7.21.

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