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A topos is defined to be a category that's equivalent to the category of sheaves on a site. Morphisms between topoi is defined by a pair of adjoint functors that behave like pull-back/push-forward of sheaves. But I was told one of the cool thing about topos is that sometimes there are morphisms of topos that are not from morphisms of a site. When people talk about this they mention the word "crystalline"...

But is there a toy example I can play around with? What's the easiest example of this?

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Actually such a thing is known as a Grothendieck or sheaf topos; there are elementary topoi which are not equivalent to a category of sheaves on a site. –  David Roberts Oct 31 '11 at 3:12
    
And I added some tags. –  David Roberts Oct 31 '11 at 3:13
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@David: I haven't conducted polls or anything but my impression is that most non-category theorists don't use the term "Grothendieck topos", saying just "topos" (and they usually don't talk about elementary toposes at all :) –  Omar Antolín-Camarena Nov 1 '11 at 1:42
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I would say that 'topos' means 'elementary topos' rather than 'sheaf topos' by default. –  Neil Strickland Apr 17 '12 at 16:13
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2 Answers 2

Let $X$ be a scheme. Let $S$ be the site of open subschemes with the Zariski topology, and let $S'$ be the site of open affine subschemes with the Zariski topology. Let $T$ and $T'$ be the associated toposes. Let $f\colon T\to T'$ be the topos map where $f^*(U)=U$ for any affine open subscheme $U$. Then $f$ is an equivalence because open affines form a base for the topology. Let $g$ be its inverse. Then $g^*$ does not restrict to a map of sites: If $V$ is an open subscheme, then $g^*(V)$ is the sheaf "represented by $V$" (i.e. it sends an affine open $U$ to $\mathrm{Hom}_X(U,V)$), but if $V$ is not affine, then $g^*(V)$ is not represented by an object in $S'$.

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+1 - I guess one could also take $\mathbb{R}^n$ and the sites of open subsets and open balls. –  Yosemite Sam Oct 31 '11 at 11:48
    
Is it true that for every (Grothendieck) topos morphism $f:X\to Y$ you can choose sites $S_X$ and $S_Y$, with $X=Sh(S_X)$ and $Y=Sh(S_Y)$, such that $f$ comes from a morphism of sites $\varphi: S_X\to S_Y$ ? –  Qfwfq Oct 31 '11 at 18:11
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If the above is true, it reminds me of the fact that you can present (nice) topological spaces $X,Y$ as simplicial complexes $S_X,S_Y$ but if the latter are too "coarse" there may be maps $f:X\to Y$ that are not realized (up to homotopy) by simplicial maps $S_X\to S_Y$; nevertheless you can always present $X$ and $Y$ by "finer" simplicial complexes such that a simplicial approximation to $f$ does exist. –  Qfwfq Oct 31 '11 at 18:16
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Qfwfq: Yes, it is. In fact, for any geometric morphism $f:X\to Y$, we can pick any site $S_Y$ generating $Y$, and then there exists a site $S_X$ generating $X$ for which $f$ is induced by a map of sites $S_Y\to S_X$. –  Mike Shulman Nov 1 '11 at 4:07
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As for you word "crystalline" it comes from the Crystalline Topos. In the book "Notes on crystalline cohomology" by Berthelot and Ogus, they show that a morphism $X\to X'$ of schemes (over a fixed base $S$) induces a morphism of the associated crystalline topoi althugh there is no morphism of the corresponding sites. This is discussed in Section 5, page 5.1.

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