The Stacks project

Lemma 82.6.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then there are commutative diagrams of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{fppf}) \ar[rr]_{f_{big, fppf}} \ar[d]_{\epsilon _ X} & & \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{fppf}) \ar[d]^{\epsilon _ Y} \\ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{\acute{e}tale}) \ar[rr]^{f_{big, {\acute{e}tale}}} & & \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{\acute{e}tale}) } \]

and

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{fppf}) \ar[rr]_{f_{big, fppf}} \ar[d]_{a_ X} & & \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{fppf}) \ar[d]^{a_ Y} \\ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \ar[rr]^{f_{small}} & & \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) } \]

with $a_ X = \pi _ X \circ \epsilon _ X$ and $a_ Y = \pi _ X \circ \epsilon _ X$.

Proof. This follows immediately from working out the definitions of the morphisms involved, see Topologies on Spaces, Section 71.7 and Section 82.5. $\square$


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