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In topos theory, there are many generalizations of topological concepts. For example, open, closed, proper and etale morphisms between toposes. However, there are also such analogous concepts in algebraic geometry.

My question is that do these concepts actually coincide? I mean, for example, a proper morphism of schemes actually induces a proper geometric morphism between some toposes induced by schemes (like etale topos)?

Although I mensioned about only morphisms, I want to know such analogous concepts in topology and algebraic geometry which are coincide at the level of toposes.

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  • $\begingroup$ I suggest you to have a look at this short book ams.org/bookstore?fn=20&arg1=alggeom&ikey=MEMO-148-705 I haven't noticed any explicit algebraic geometry in it, but I'm sure it contains some pointers. But maybe you come from that book? :-) $\endgroup$
    – fosco
    Commented Jan 9, 2014 at 18:28
  • $\begingroup$ Thank you, but I know that book. And as you mentioned, there's no explicit descriptions. $\endgroup$
    – Fujita Tomomi
    Commented Jan 9, 2014 at 19:38
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    $\begingroup$ I think Urs' answer is interesting and a good one, but I also want to add that in some sense these really are the same thing in the sense that to any topological space one can associate a Grothendieck topology. Where the difference lies, in my mind, is in the notion of having "enough points." There is quite a bit of subtlety here though that I'm not an expert in. I recommend looking at locales however. A good place to start might be ncatlab.org/nlab/show/locale. $\endgroup$
    – Jonathan Beardsley
    Commented Jan 10, 2014 at 1:21
  • $\begingroup$ In 2021 fosco's comment's link is broken. A current link is bookstore.ams.org/memo-148-705, to Proper Maps of Toposes by I. Moerdijk and J.J.C. Vermeulen, Memoirs of AMS, 2000. $\endgroup$
    – Zach Teitler
    Commented Nov 4, 2021 at 17:21

2 Answers 2

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I suppose it's true that this is an aspect that deserves to receive more attention.

One place where algebraic geometry is systematically done via the topos theory of the étale toposes of the given spaces is Jacob Lurie's "Structured Spaces" and generally the "E-∞ Geometry" based on that.

(If you don't want to get into higher geometry you can simply ignore all the $\infty$-prefixes there and still get a discussion of the foundations of algebraic geometry that is conceptually cleaner than most of what one sees offered elsewhere).

For instance one statement of the kind that you are looking for is theorem 1.2.1 in "Quasi-Coherent Sheaves and Tannaka Duality Theorems". This says that a morphism of affine schemes is étale precisely if the induced map on étale toposes is an étale geometric morphism of ringed toposes.

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  • $\begingroup$ Where is the definition of "etale geometric morphism of ringed toposes" to be found? If one removes the $\infty$'s, is the above Theorem 1.2.1 different from the functorial characterization of etaleness for maps of schemes? (That is, does it tell us something different from EGA, or with a different method?) I do not know what a "proper" geometric map of toposes is (where is the definition given?), but should techniques that address an "etale" version be relevant to properties such as properness? Sorry for the list of questions. $\endgroup$
    – user76758
    Commented Jan 10, 2014 at 8:54
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    $\begingroup$ An etale geometric morphism of ringed toposes (X,O_X) --> (Y,O_Y) is an etale geometric morphism of the underlying toposes f : X --> Y (ncatlab.org/nlab/show/etale+geometric+morphism) together with an equivalence f^* O_Y = O_X . For proper geometric morphism see ncatlab.org/nlab/show/proper+geometric+morphism . Concerning EGA I haven't checked (sorry), but probably best is to compare Monique Hakim's "Topos annelés et schémas relatifs", Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972). $\endgroup$
    – Urs Schreiber
    Commented Jan 10, 2014 at 13:09
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I found this question while I was also wondering about proper maps of schemes versus proper geometric morphisms of their small étale toposes. I will add a partial result for future reference.

I'll consider toposes of sheaves of sets (no ringed toposes).

A topos $\mathcal{E}$ is tidy or strongly compact iff the global sections functor $\Gamma : \mathcal{E} \to \mathbf{Sets}$ preserved filtered colimits, and it is proper or compact iff $\Gamma$ preserves directed unions of subterminal objects. All tidy toposes are proper.

It is shown in SGA4, Exposé VI, Exemple 1.22 that the small étale topos $X_\mathrm{\acute{e}t}$ is tidy if and only if $X$ is quasi-compact quasi-separated (and similarly for the small Zariski topos). So if $X$ is a proper scheme, then $X_\mathrm{\acute{e}t}$ is a tidy (in particular, proper) topos. But if $X$ is an affine scheme, then the associated topos $X_\mathrm{\acute{e}t}$ is proper as well.

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