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There is a general notion of a geometric point in topos theory. A geometric point is a geometric morphism $Set\to T$.

There is also a notion of a geometric point in algebraic geometry. A geometric point is a point $k\to S$ where $k$ is algebraically closed.

Why do these notions agree?

The geometric points of the ├ętale site of a scheme are maps $k\to S$ where $k$ is separably closed. Again, why are these the same as geometric morphisms from the category of sets into the ├ętale topos?

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Obviously by Zariski and étale toposes, I mean the appropriate slicey-guys over a scheme. – Harry Gindi Dec 10 '10 at 6:44
The geom. pts of topos of a sober top. space (e.g., scheme, loc. Hausdorff space), are (up to equiv.) stalk functors at pts of the top. space, with distinct physical pts giving inequiv. functors. (For a scheme $S$ can use pullback along Spec($\overline{k(s)}) \rightarrow S$, but silly since Spec($\overline{k(s)}) \rightarrow {\rm{Spec}}(k(s))$ induces equiv. of Zariski topoi.) This is proved in "Sheaves in Geom. & Logic" and near end of 2nd volume of SGA4, where Grothendieck proves that geom. pts of etale topos of $S$ are (up to equiv.) pullbacks along Spec($k(s)_{\rm{sep}}) \rightarrow S$. – BCnrd Dec 10 '10 at 10:04
Thanks for the answer and the references! – Harry Gindi Dec 10 '10 at 18:36

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