Let S be a scheme. Let (Sch/S) be a small category of schemes over S (including essentially all finitely presented schemes affine over S). Let E = (Sch/S)zar denote the gros Zariski topos with its local ring object A1.

Is there a nice way to construct the petit Zariski topos X = Szar out of the locally ringed topos E? (By nice I mean, for example, that there is a universal property that the locally ringed topos X possesses with respect to E.)

There are variations of this question in which I am also interested: For example, one can replace E by the gros étale (or fppf or fpqc) topos (Sch/S)ét and ask for the construction of Szar out of (Sch/S)ét. Or one can replace X by the petit étale (or fppf or fpqc) topos Sét and ask for the construction of it out of E = (Sch/S)zar.

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    $\begingroup$ There was quite a long discussion on this topic over at the n-Category Cafe, here: golem.ph.utexas.edu/category/2009/01/… Have you seen it? $\endgroup$ – Manny Reyes Apr 16 '10 at 22:54
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    $\begingroup$ @Manny: Thanks for pointing out that discussion to me. Denis-Charles Cisinski's comments there were quite helpful to me. Furthermore, the discussion contains a link to a paper by Mathieu Anel, "Grothendieck topologies from unique factorisation systems" where my question is completely answered in the case that my scheme S is affine, i.e. Spec A: E classifies local rings in topoi while X classifies local rings that are localisations of A, so X is the subtopos of those objects Y of E such that A^1 restricted to Y is a localisation of A. This should generalise easily to general schemes S. $\endgroup$ – Marc Nieper-Wißkirchen Nov 19 '10 at 11:20
  • $\begingroup$ The gros topos $E$ is the classifying topos for local rings. So maybe the $2$-category $C$ of toposes with maps into $E$ is like the category of locally ringed spaces. Then maybe $\text{Spec} : \text{Ring} \rightarrow C$ is adjoint to global sections. $\endgroup$ – Dean Young Apr 30 '19 at 17:24
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    $\begingroup$ @Dean: The morphisms in $C$ are not what we might expect: Their ring-theoretic parts are required to be isomorphisms instead of local homomorphisms (as would be the case in the category of locally ringed spaces). The Spec functor is indeed an adjoint, if we let it map to the category of locally ringed toposes (objects are pairs $(\mathcal{E},\mathcal{O}_\mathcal{E})$, morphisms are pairs $(f:\mathcal{E}\to\mathcal{F}, f^\sharp:f^{-1}\mathcal{O}_\mathcal{F}\to\mathcal{O}_\mathcal{E})$), see for instance Sect. 12 of these notes. $\endgroup$ – Ingo Blechschmidt May 1 '19 at 15:34

Many of these toposes admit descriptions as internal classifying toposes, hence indeed enjoy useful universal properties. Here is a selection of such descriptions:

Constructing the big Zariski topos from the little Zariski topos. The big Zariski topos of a scheme $S$ is the externalization of the result of constructing, internally to the little Zariski topos of $S$, the classifying topos of local $\mathcal{O}_S$-algebras which are local over $\mathcal{O}_S$. (One might hope that it would simply be the internal big Zariski topos of $\mathcal{O}_S$, that is the classifying topos of local $\mathcal{O}_S$-algebras where the structure morphism needn't be local. This alternative description is true if $S$ is of dimension $0$.)

Constructing the little Zariski topos from the big Zariski topos. Additionally to $\mathbf{A}^1$, which is the functor $T \mapsto \Gamma(T,\mathcal{O}_T)$, the big Zariski topos contains an additional local ring object: The functor $\flat \mathbf{A}^1$, which maps an $S$-scheme $(f : T \to S)$ to $\Gamma(T,f^{-1}\mathcal{O}_S)$. There is a local ring homomorphism $\flat \mathbf{A}^1 \to \mathbf{A}^1$, and the little Zariski topos can be characterized as the largest subtopos of the big Zariski topos where this morphism is an isomorphism. (This fits with the comments as follows. If $S = \operatorname{Spec}(A)$ is affine, we have the string of ring homomorphisms $\underline{A} \to \flat \mathbf{A}^1 \to \mathbf{A}^1$ starting in the constant sheaf $\underline{A}$. The map $\underline{A} \to \flat \mathbf{A}^1$ is always a localization, and the composition is iff $\flat \mathbf{A}^1 \to \mathbf{A}^1$ is an isomorphism.)

Constructing the big étale topos from the big Zariski topos. The big étale topos of a scheme $S$ is the largest subtopos of the big Zariski topos of $S$ where $\mathbf{A}^1$ is separably closed. This fact is essentially a restatement of Gavin Wraith's theorem on what the big étale topos classifies.

Constructing the big infinitesimal topos from the little Zariski topos. A back-of-the-envelope computation indicates that the recent result of Matthias Hutzler on what the infinitesimal topos of an affine scheme classifies can be relativized to the non-affine case as follows. The big infinitesimal topos of a scheme $S$ is the externalization of constructing, internally to the little Zariski topos of $S$, the classifying topos of local and local-over-$\mathcal{O}_S$ $\mathcal{O}_S$-algebras equipped with a nilpotent ideal.

Some details can be found in Sections 12 and 21 of these notes.

A word of warning: When I say "the largest subtopos where foo", I refer to the largest element in the poset of subtoposes which validate foo from their internal language. (By general abstract nonsense (more or less the existence of classifying toposes), such a largest element always exists in case foo is a set-indexed conjunction of geometric implications.) In particular, I'm not referring to "the subtopos of those objects $Y$ of $E$ such that $\mathbf{A}^1$ restricted to $Y$ enjoys foo" (as in the comments). Indeed, this category is in general not a subtopos (typically it doesn't contain the terminal object). Maybe I interpreted that phrase too literally.

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  • $\begingroup$ That's nice! Do you know characterizations of some other sheaves in a similar way? Fpqc, fppf, Nisnevich...? $\endgroup$ – მამუკა ჯიბლაძე May 1 '19 at 19:36
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    $\begingroup$ The fpqc and fppf topologies coincide in the case that the site only contains schemes which are (locally) of finite presentation. Without this finiteness condition, not even the big Zariski topos will classify any reasonable theory. That said, the big fppf topos classifies fppf-local rings (Definition 21.4 in the linked notes). Conjecturally, these are the same as algebraically closed local rings. Hence the big fppf topos is the largest subtopos of the big Zariski topos where $\mathbf{A}^1$ is fppf-local. The answer for the Nisnevich topology is to the best of my knowledge unknown. $\endgroup$ – Ingo Blechschmidt May 2 '19 at 11:15

I will deal with étale toposes because they behave much better in every possible way. They are also easier to define, althought they require substantially more commutative algebra to work with in practice:

The gros étale topos for $S$ is just $((Shv(Aff_{\acute Et})\downarrow S).$ We can construct from it the petit étale topos by considering $Shv(\acute Et \downarrow S)$, where $(\acute Et \downarrow S)$ is the subcategory of the gros étale topos consisting of étale morphisms $A\to S$ where $A$ is affine. This site is equipped with the induced topology.

Now for the ring object. For the petit topos, we let $\mathcal{O}_S$ be defined simply the sheaf sending any affine scheme to its corresponding ring (exercise: Show that this is a sheaf). This defines a ring object in the category of sheaves on the small site (exercise: Prove this. (Hint: Think of the definition of a group object and recall that the Yoneda embedding is full.)). For the large topos, we just let it be the base change of the affine line. It's not hard to show that they agree on étale morphisms $A\to S$ for A affine.

It turns out that the gros and petit toposes have a geometric morphism induced by the inclusion of the small site into the large site. I don't know if there is a specific universal property, per se, but it turns out that they are "homotopy equivalent" in a suitable sense.

For an explanation of the homotopy condition, see

Mac Lane and Moerdijk - Sheaves in Geometry and Logic Chapter 7.

Edit: If I remember correctly, the statement about "homotopy equivalence" does not work in the fppf or fpqc topologies. The small flat sites are too small, in some sense.

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  • $\begingroup$ Harry, thanks for your answer. However, it is not the answer I was looking for. It is clear that one can construct the petit (étale) topos out of the gros (étale) topos and that the ring object can be constructed the way you described it. But the question was whether this construction can be viewed as a natural one or whether the geometric morphisms between the gros and the petit toposes fulfill some universal properties. (For example, if I am not mistaken, constructing the petit étale topos out of the petit Zariski topos has a nice description; see M. Hakim's thesis.) Marc $\endgroup$ – Marc Nieper-Wißkirchen Apr 13 '10 at 9:11
  • $\begingroup$ Where in M. Hakim's thesis? $\endgroup$ – Harry Gindi Apr 13 '10 at 10:13
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    $\begingroup$ See for example in section IV.5 of her thesis. A good account on what it says about the étale spectrum can be found here: springerlink.com/content/10x9002103602132/fulltext.pdf $\endgroup$ – Marc Nieper-Wißkirchen Apr 16 '10 at 15:24
  • $\begingroup$ I was able to find her thesis in one of the usual places =). $\endgroup$ – Harry Gindi Apr 16 '10 at 15:37

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