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January 22, 2009

Petit Topos, Gros Topos

Posted by John Baez

As I struggle to learn algebraic geometry, I’m running into lots of questions, and I hope some of these will be fun for readers of this blog.

Here’s one: What’s the relation between the topos of sheaves on the little Zariski site and the topos of sheaves on the big Zariski site?

In this marvelous age, anyone can read about the ‘little Zariski site’ and the ‘big Zariski site’ on Wikipedia. But let me say a word or two to those who are blissfully ignorant of such things — before asking some questions that only those familiar with such things will enjoy.

A sheaf on a topological space gives, for each open set in that space, a set of ‘sections’. You can restrict a section from a big open set to a smaller one. You can also glue together a bunch of sections on little open sets U iU_i to get a section on an open set that they cover, at least if these sections agree on the intersections U iU jU_i \cap U_j.

Now, open sets form a poset, which is a very simple kind of category. More generally, for any category, a Grothendieck topology gives a concept of when a bunch of objects ‘cover’ another object. So, we can define a sheaf on any category with a Grothendieck topology.

(In fact, just as we can specify a topology on a space by giving a basis of open sets, it’s often handy to specify a Grothendieck topology by giving something called a Grothendieck ‘pretopology’ — and that’s what we’ll do below.)

A category with a Grothendieck topology is also called a ‘site’. So, we can define sheaves on any site! And, the category of all sheaves on a site is very nice: it’s a topos.

But my questions are about the topos of sheaves on the site of schemes.

A scheme is a kind of space that generalizes an algebraic variety. An affine scheme is just a commutative ring in disguise: we can think of any commutative ring as the ring of ‘functions’ on some space, and we call that space an ‘affine scheme’. We can build more general schemes by gluing together affine schemes, much as we build manifolds by gluing together coordinate charts.

To define the little Zariski site of a scheme XX, we think of it as a topological space and take its poset of open subsets, O(X)O(X). Then we give this category O(X)O(X) the Grothendieck pretopology where a covering familly is a jointly surjective family of open immersions. (Such a family is close to being an ‘open cover’ in the most naive sense.) This makes O(X)O(X) into a site — the little Zariski site of XX.

To define the big Zariski site of a scheme XX, we take the category of schemes, say SchSch. Then we give this category a pretopology where a covering family is a jointly surjective family of scheme-theoretic open immersions. This makes SchSch into a site. The category of schemes over XX, say Sch/XSch/X, then becomes a site in its own right — the big Zariski site of XX.

(I’m not too clear on the importance of the extra qualifier scheme-theoretic. Wikipedia discusses it… but I somehow doubt that this is what I’m mainly interested in right now. I’m just trying to get a grip on the basic idea of turning a scheme into a topos in two ways, ‘small’ and ‘big’.)

The topos of sheaves on the little Zariski site of XX is called the petit topos while the sheaves on the big Zariski site of XX is called the gros topos.

Here are a bunch of questions:

  • Do the petit and gros topos of XX fail to be equivalent as categories? Perhaps this already fails when XX is a point. Maybe the petit topos is localic and the gros one is not? Mac Lane and Moerdijk say on page 414 of Sheaves in Geometry and Logic that they’re homotopy equivalent — presumably this is a kind of fallback position.
  • Is the (2-)functor assigning to each scheme its petit topos full and faithful, viewed as a (2-)functor from SchSch to the (2-)category of locally ringed topoi? How about the functor assigning to each scheme its gros topos?
  • What is the petit topos of XX the classifying topos for? How about the gros topos?

I probably have a lot more questions, but that’s a start… Feel free to tell me stuff I should know, but didn’t ask — as long as you keep it simple!

Posted at January 22, 2009 12:30 AM UTC

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29 Comments & 1 Trackback

Re: Petit Topos, Gros Topos

I’ve thought a bit about these things in the past couple of years. I’m not at all an expert, but there are so few experts on these things, I might as well say something.

The way I think about the big and small toposes associated to XX (I’ll stick to the English terms) is that they’re both worlds of spaces over XX, though in the small topos, the spaces are so similar to XX that you almost might as well be talking about XX. So you can be forgiven for thinking about a small topos as a single space, but it seems misleading to think about a big topos that way.

1. Yes, the big and small toposes are not equivalent in the case of the point. (Maybe not ever for nonempty schemes?) The small topos over the point is just the category of sets, where every epimorphism has a section. (If you don’t like choice, look at epimorphisms of compact objects.) This is not true in the big topos. (Take a scheme and an open cover with no section.) There’s probably a better argument, especially if you only care about showing the obvious map between the two toposes is not an equivalence, a less evil question.

2. The small-topos functor is not faithful for the simple reason that it doesn’t see nilpotents. So there are lots of ring endomorphisms of k[x]/(x 2)k[x]/(x^2) but they are all the identity on the small topos.

It’s also not full. Let kk be a field, and let AA be a ring with no maps from kk. Then there are no maps from Spec AA to Spec kk. But Spec kk is just the point, and the topos of Spec AA, like any topos, maps to the point topos.

I’d have to think a bit about the other questions.

Posted by: James on January 22, 2009 5:50 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

This is a good question and not one that I understand either; I hope that there is someone out there who can enlighten us. I just want to point out two things.

Firstly, if anyone out there doesn’t happen to be comfortable with schemes, there is the same distinction for ordinary spaces. For a topological space XX we can consider sheaves on its ordinary site of open sets, givin a petit topos. Or we can consider sheaves on Top/XTop/X, giving a gros topos.

Somewhere in between, we can consider sheaves on LH/XLH/X, where LHLH is the category of topological spaces and local homeomorphisms. It turns out that LH/XLH/X is actually equivalent to the petit topos of XX, and therefore so is the category of sheaves on it; so this just reconstructs the petit topos. I don’t know whether this point is helpful; it probably doesn’t have a scheme-theoretic analogue.

Secondly, there are also set-theoretic issues. In order for the category of sheaves on a site to be a topos, the site has to be small (or, at least, have a small “dense” subcategory). But Sch/XSch/X and Top/XTop/X are not small. Of course you can just restrict to some small full subcategory of them, or (what amounts to the same thing) assume a Grothendieck universe, but this is something that has always gotten in the way of my understanding what the gros topos is supposed to represent.

Posted by: Mike Shulman on January 22, 2009 8:49 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

For a topological space XX we can consider sheaves on its ordinary site of open sets, givin a petit topos. Or we can consider sheaves on Top/XTop/X, giving a gros topos.

A related question:

given a sheaf VV on TopXTop\downarrow X, we can consider the corresponding fibration regarded as a topological space PXP \to X. Then there is the sheaf V¯\bar V on all of TopTop which is represented by PP.

I am wondering: shouldn’t this be the left Kan extension of VV along the forgetful functor TopXTopTop\downarrow X \to Top, i.e. the functor

V¯():= UXTopXTop(,U)V(UX) \bar V(-) := \int^{U \to X \in Top\downarrow X} Top(-,U) \cdot V(U \to X) ?

Posted by: Urs Schreiber on January 22, 2009 10:12 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

Have a look at the discussion in Metzler’s paper on stacks starting on p. 41. While it may not completely answer your questions, it has some useful references. I keep meaning to think about these things but other stuff intervenes, so I am speaking from ignorance.

Posted by: Eugene Lerman on January 22, 2009 10:47 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

There are two functors relating the categories of sheaves on the small and the big site.

First if we have a sheaf on the Big Site Sch/X we can simply restrict it to the open sets and get a sheaf there (I didn’t think too long about open immersions, but I hope this is right).

On the other hand, starting with a sheaf F on the little Zariski site we can pull it back along arbitrary mappings Y → X. Now we can take the global sections as the value of the “big” sheaf on Y → X.

As we see immediately if we start with a sheaf on the little site we don’t change anything on open subsets U → X. So extending first and then restricting gives the identity.

If we start with a big sheaf we cannot expect to keep the full information if we restrict it, because there are many maps that are far from being a submersion. So there are much more sheaves on the big site than on the little.

Posted by: Thomas Nikolaus on January 22, 2009 11:18 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

If Sch/XSch/X denotes the category of schemes (locally) of finite presentation over XX endowed with the Zariski topology, the category of sheaves on Sch/XSch/X and the category of sheaves on the small Zariski site of XX are very different by nature: the representable sheaves on the gros topos are all the schemes (locally) of finite presentation over XX, while the representable sheaves on the petit topos are the open subschemes of XX.

To express further the differences between the petit and the gros topos of XX, we can notice that the Yoneda embedding gives a fully faithful functor from Sch/XSch/X to the category of sheaves on the gros topos. In particular, XX is completely determined by its gros topos; however, the scheme XX is not determined by its petit topos: the petit topos only determines the underlying topological space of XX. To determine XX, you need to know its petit topos as well as its sheaf of rings O XO_X. And the data of this sheaf of rings is a kind of disguised link to a gros topos! Indeed, let us consider first the case where X=Spec(Z)X=Spec(\mathbf{Z}). Then the gros Zariski topos of Spec(Z)Spec(\mathbf{Z}) is the classifying topos for local rings: given a topos TT, a sheaf of local rings on TT (I mean by this a sheaf of rings OO which is locally a local ring, i.e., in the case where the topos has enough points, for any point xx of the topos TT, the ring O xO_x is a local ring) is the same thing as a (geometric) morphism of topoi from TT to the big Zariski topos of Spec(Z)Spec(\mathbf{Z}). The universal local ring is the Zariski sheaf represented by the affine line on Sch/Spec(Z)Sch/Spec(\mathbf{Z}).

The data of the sheaf of rings O XO_X thus gives you the relation of XX with all the schemes of finite presentation over Spec(Z)Spec(\mathbf{Z}). As any ring is a filtered colimit of rings of finite presentation, you can understand from this all the sets Hom(X,Y)Hom(X,Y) for any scheme YY, hence recover XX completely by the Yoneda Lemma (this is why the definitions of schemes as locally ringed spaces or as sheaves on the gros topos lead to equivalent notions after all).

There is a systematic ‘toposic’ way to recover the gros Zariski topos of a scheme XX from the category Sch/XSch/X and from the petits topoi of schemes (locally) of finite presentation over XX. Indeed, we have a (pseudo-)functor () Zar:Sch/XTopoi(-)_{Zar}: Sch/X\to Topoi which associates to a scheme YY over XX its petit topos Y ZarY_{Zar}. Then the gros topos of XX can be described as the lax 22-colimit of this pseudo-functor. In particular, the canonical morphisms of topoi from Y ZarY_{Zar} to the gros Zariski topos of XX form the universal ‘lax cocone’.

This is a general process which works for some other different Grothendieck topologies on Sch/XSch/X. For instance, one might be interested by henselian local rings (a commutative ring with unit RR is henselian if, any unital polynomial with coefficients in RR which has a root in the residue field of RR has a root in RR). The Nisnevich gros topos of Spec(Z)Spec(\mathbf{Z}) (i.e. the topos of Nisnevich sheaves on Sch/XSch/X) is the classifying topos for henselian local rings. We can also recover the gros Nisnevich topos of XX as lax 22-colimit of the small Nisnevich topoi of the schemes of finite presentation over XX. Similarly, we have such a picture for the étale topology; it corresponds in a similar way to the notion of strict henselian ring (i.e. henselian local ring with separably closed residue field).

We can also change the size condition I used (e.g. replace ‘of finite presentation’ by of ‘α\alpha-small presentation’ for some given cardinal α\alpha, which might even look like no size condition at all if α\alpha is the cardinal of some universe; this corresponds then to the très très gros topos).

Posted by: Denis-Charles Cisinski on January 23, 2009 1:41 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

I feel like I would understand the gros topos better if I knew the answer to John’s last question: what is it the classifying topos of? You answered that question for X=SpecZX=Spec Z: local rings. So I can think of the gros topos Sh(Sch/SpecZ)Sh(Sch/Spec Z) as “the generalized-space of all local rings.” Is there a corresponding description for other gros toposes?

Also, is there a version of this in the purely topological case?

Posted by: Mike Shulman on January 23, 2009 4:17 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

A few comments on the comments above:

1. Nikolaus’s post: There are actually three functors relating the big and small toposes. More precisely, there is a topos map from the big topos to the small topos, and this map has a section ff. Then not only does f *f^* have a right adjoint f *f_* but it also has a left adjoint f !f_!.

2. Cisinski’s first paragraph: It might be worth pointing out that while it may be true that a sheaf on the small site is representable if and only if it’s an open subscheme, this is because the site you’re using is the category of open subschemes. But there are lots of sheaves on the small site that are representable by other schemes. For instance, let i:0A 1i:0\to A^1 be the closed immersion from the point into the affine line, and let FF be the two-point sheaf on the point. Then i *(F)i_*(F) is representable by the non-separated scheme formed by gluing two copies of the affine line together everywhere but at the point 00. I would actually expect that every sheaf on the small site is represented by some scheme, usually incredibly non-separated. This is just because every sheaf is a colimit of representables, and you can always glue schemes together along open subschemes. Though, maybe I’ve overlooked something.

In other words, I think the category LH/XLH/X of Mike Shulman’s does have a scheme-theoretic analogue, though no one ever uses it.

Also, I don’t think it’s necessary to have any finiteness conditions on the objects of the big Zariski site, but I don’t think it hurts anything here either. (Well, it does let you be free from set-theoretic worries.)

One question I have is about the statement that a scheme “is completely determined by its gros topos”. What exactly do you mean by this?

3. The second paragraph raises a point I agree with and I think is worth repeating (as I often do on this blog). The best way of setting up algebraic geometry is as a subcategory of the big topos (over whatever base you want, usually Spec ZZ). Then all the gluing constructions needed to patch affine schemes together work by topos-theoretic generalities. This approach is a bit different than the one that it looked like John was suggesting above. Here we have do everything inside of a single big topos and then schemes are identified with the objects of that topos they represent (in fact, I prefer to define them that way). In the other approach, a scheme is identified with its big topos and then algebraic geometry lives inside of some category of toposes.

In fact, I like this approach so much, I think people should do it in pretty much every kind of geometry, for instance diffeology, but I haven’t been able to convince anyone else of that.

4. Shulman made the point about set-theoretic issues. I think these ought to be orthogonal to the spirit of sites and toposes, but I also think that they’re not completely ironed out yet. Grothendieck’s approach was to use universes, but I believe deep in my soul that that’s cheating. On the other hand, something has to be done because as Shulman points out the category of sheaves on the big site doesn’t have small hom sets. One approach is not to consider all sheaves but only those that are a small colimit of representable sheaves. There is something about this approach that really smells right to me, though I can’t say I know that it will work. In other words, if we redefine a (Grothendieck) topos to be a category equivalent to the category of sheaves of sets on a site which are isomorphic to small colimits of representables, do we loose anything important?

In the example of scheme theory (as opposed to large sites in the abstract), maybe Cisinski’s approach of restricting the site to things locally of finite presentation works, but I haven’t been convinced of that yet.

5. Last, I have no idea if it’s possible to say what the big topos is a classifying space of in an enlightening way. But then again, I’ve never found the point of view of classifying toposes enlightening. This might say more about me than the topic, though. I find it hard to read anything on toposes since Grothendieck, perhaps because the authors are motivated more by logic (it seems to me) and less by geometry.

Posted by: James on January 24, 2009 5:13 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

In other words, if we redefine a (Grothendieck) topos to be a category equivalent to the category of sheaves of sets on a site which are isomorphic to small colimits of representables, do we lose anything important?

Well, it won’t be an elementary topos any more. It probably won’t even be cartesian closed. And although it will be locally small, it won’t in general be well-powered. But of course the real answer depends on your definition of “important.” (-:

Also, in terms of foundations, you actually still need universes or some such-like to define the category you propose, although once you’ve defined it it turns out to be locally small.

Posted by: Mike Shulman on January 24, 2009 9:58 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

If you could come up with an example showing that it can fail to be cartesian closed, I’d be interested. That’s a pretty important property. I don’t think I care too much about well-poweredness.

As for the foundations, do you really need universes? I would think that instead of first building the whole sheaf category and then restricting to the subcategory that I mentioned, you could build it up directly by looking at small diagrams of representables.

This actually reminds me of the question What is the right definition of a coherent R-module (or sheaf)? The two most common definitions, (1) that the module be finitely generated and (2) that the module be finitely presented, each have problems. I wouldn’t be surprised if the best definition is that the module should have a finite resolution by finitely generated modules. Perhaps doing an analogous thing for sheaves of sets is the right idea. For instance, we might try taking sheaves which have finite simplicial resolutions by sheaves which are small coproducts of representables.

Posted by: James on January 26, 2009 1:56 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

Well, consider just the case of presheaves (sheaves on a trivial site). According to Limits of small functors, we have for instance:

  • If KK is complete, then so is the category PKP K of small presheaves on KK (“small” means “a small colimit of representables”)
  • If KK is cartesian closed, so is PKP K.

However, if KK is large then its lack of these properties can carry over to PKP K. For instance, if KK is a large discrete category, then PKP K has no terminal object. So you probably wouldn’t want a definition as general as “the category of small sheaves on an arbitrary site.” They also give a necessary and sufficient condition on KK for PKP K to be (cartesian) closed, although I haven’t thought of an easily provable counterexample yet.

Posted by: Mike Shulman on January 27, 2009 5:34 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

The notion of “accessible category” seems very relevant here. A category AA is accessible if (roughly) every object is a κ\kappa-filtered colimit of some fixed (small) set of objects; κ\kappa is a big fixed cardinal. An accessible presheaf of sets on AA is a small colimit of representables; it turns out that the category of accessible presheaves of sets is complete and cocomplete (see Makkai and Pare, “Accessible Categories” (CM104), especially Prop 2.4.2 and Cor 2.4.6).

Unfortunately, the notion of accessible category isn’t quite good enough: not every small category is accessible. How does one characterize the class of (large) categories CC with the property that the category of accessible presheaves of sets on CC is complete and cocomplete?

Posted by: Charles Rezk on January 27, 2009 5:13 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

Any small category with split idempotents is accessible, though, and splitting of idempotents doesn’t change the category of presheaves (since Set has split idempotents and idempotent-splittings are absolute colimits). So at least as far as (pre)sheaves are concerned, it doesn’t matter that not all small categories are accessible.

So perhaps we should consider categories of accessible sheaves on accessible sites?

Posted by: Mike Shulman on January 27, 2009 7:29 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

Except, of course, that TopTop is not accessible, and probably SchSch isn’t either.

Posted by: Mike Shulman on January 27, 2009 10:14 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

I suspect that variants of Top, like combinatorially generated spaces, are accessible. That’s a good point about split idempotents.

Posted by: Charles Rezk on January 28, 2009 12:00 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

Well, yes, of course any locally presentable category, like Delta-generated spaces, is accessible. But it’s not obvious to me that a replacement of that sort is justified in the contexts where one might be interested in sheaves on TopTop. In topos theory, for instance, one really cares about the category of locales, which definitely is not accessible and which I would bet cannot be easily modified to become so.

Likewise, the category of affine schemes is definitely not accessible (since its opposite is locally presentable), and it seems doubtful to me that it has any accessible replacement.

Posted by: Mike Shulman on January 28, 2009 3:18 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

I think you’re right that you can build it up with small diagrams of representables, though.

Posted by: Mike Shulman on January 27, 2009 2:15 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

The notion of representable sheaf is not canonically defined in a topos. What I mean by representable sheaf depends on the choice of a site! Otherwise, the most canonical site for a topos TT is the category TT itself: TT is equivalent to the category of sheaves on TT for the canonical topology on TT. When I stressed that the representable sheaves on the small Zariski topos of a scheme XX are the open subsets, I just wanted to insist that the small topos only characterizes the underlying topological space of XX: if you forget the structural sheaf of local rings on XX, then you forget a huge part of the geometry of XX: there are plenty non isomorphic schemes which have equivalent small Zariski topoi. For instance the small Zarsiki topos of the spectrum of discrete valuation ring is always equivalent to the category of arrows of SetsSets (the underlying topological space has only two points: a closed one, and open one).

About the size condition: the classifying topos for local rings is really the gros topos of Zarsiki sheaves on the site of schemes of finite presentation over Spec(Z)Spec(Z). If we consider “bigger gros topoi”, I don’t know what we are classifying then…

My assertion that a scheme is completely determined by its gros topos was very clumsy. To keep track of the geometry, you always need to remember what the affine line is (otherwise, I don’t know how to define vector bundles, how to describe nicely smooth maps…), which is similar to keep track of the structural ring after all… I guess that what I had in mind is that, if you remember the Yoneda embedding of Sch/XSch/X in the gros Zariski topose, then the terminal sheaf gives you the functor which represents XX in Sch/XSch/X, which is quite a tautology. However, if you do the same with the small topos, you will only get the poset of open subsets of XX

Otherwise I srongly agree with you that the gros topos of differential manifolds is a wonderful category to work with. This is a very nice completion of the category of smooth differential manifolds by colimits, and it allows plenty of nice constructions. For instance, one can endow this gros topos with a nice (i.e. combinatorial proper) model category structure which is Quillen equivalent to the usual model category structure on topological spaces. This means that, in some sense, any homotopy type is locally a smooth manifold. This last assertion is just a fancy reformulation that any homotopy type is locally contractible, but, this allows for instance to think of de Rham cohomology as a very nice cohomology which is canonically defined on all homotopy types… I just would like to mention that people who are interested by the homotopy theory of schemes or by derived algebraic geometry are aware of this point of view, and sometimes even use it.

Posted by: Denis-Charles Cisinski on January 24, 2009 10:58 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

OK. I think we agree about everything. In particular, I didn’t think you thought representability was independent of the site. I just thought there was a chance someone else might think that.

Posted by: James on January 26, 2009 12:30 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

Denis wrote:

We can also recover the gros Nisnevich topos of X as lax 2-colimit of the small Nisnevich topoi of the schemes of finite presentation over X.

Do you really mean lax 2-colimit as opposed to pseudo 2-colimit? I’m wondering because I seem be running into a description of the gros Zariski topos of Spec(R)Spec(R) as a kind of pullback, and it seems like it may be a lax instead of a pseudo pullback, which seems a bit surprising somehow.

By the way, I should thank your for your incredibly helpful posts on this thread! I apologize for not responding to them sooner. I’ll eventually have a lot more to say (and ask) about these subjects.

Posted by: John Baez on February 11, 2009 6:39 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

Does anyone know of sufficient conditions for the category of small presheaves on a large category to be an elementary topos?

Posted by: Richard Garner on January 29, 2009 3:39 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

A2.1.5 in the Elephant gives the condition that each slice category is essentially small. This applies to any groupoid and also to, for example, the large poset of small ordinal numbers. Of course, it’s quite restrictive, but I don’t know of any weaker condition.

Posted by: Mike Shulman on January 29, 2009 4:29 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

It’e been a while since Richard’s question, but my talk from CT2013 gave a sufficient condition. Namely if the large site is the colimit of the right adjoint functors from an Ord-sequence of fibrations of sites, such that the induced geometric morphisms are open surjections, then if for any object in the category of small sheaves, considered as a sheaf on one of the sites in the sequence, the comparison maps between power objects eventually become isomorphisms, then the category of small sheaves is a topos. If all the geometric morphisms in this sequence are localic, then one only needs to check the power objects of those objects in the base topos.

In practice we only have as examples a large sites which are “Easton products” of small posets with the double-negation topology, coming from set theory. I’d love to get my hands on a non-Boolean example.

Posted by: David Roberts on August 9, 2013 2:10 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

Nice! I’m especially impressed that you found this 4-year-old discussion to post an answer to it.

Posted by: Mike Shulman on August 9, 2013 10:22 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

Actually it was a fluke, following a link from David Corfield on the nForum. I’m glad he can keep track of all the old conversations!

Posted by: David Roberts on August 10, 2013 12:35 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

David C has an extraordinary ability to do that.

Posted by: Tom Leinster on August 10, 2013 4:23 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

Perhaps this paper posted on the arXiv today may be helpful? I haven’t digested it yet but he seems to be presenting a general context that gives rise to “petit/gros” pairs of toposes.

Posted by: Mike Shulman on February 9, 2009 6:01 PM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

Okay, now that I’ve skimmed it, the main points of interest for this thread seem to be:

  • The petit (Zariski) topos of an affine scheme SpecRSpec R classifies local forms of RR, meaning localization homomorphisms RSR\to S where SS is local; while
  • The gros (Zariski) topos of SpecRSpec R classifies local RR-algebras (arbitrary homomorphisms RSR\to S where SS is local).

As a particular case, when R=ZR=Z, the gros topos just classifies local rings. The petit and gros toposes for other coverages (etale, proper, Nisnevich) have similar descriptions.

Posted by: Mike Shulman on February 11, 2009 3:08 AM | Permalink | Reply to this

Re: Petit Topos, Gros Topos

Thanks a lot for pointing out the existence of this new paper. It completely answers my third question!

I feel a bit sad, because Jim Dolan and I were just coming close to answering this question on our own. However, our approach is based on a different philosophy, which I believe will be interesting in its own right. We plan to put something online about this in a couple of weeks.

Posted by: John Baez on February 11, 2009 6:28 AM | Permalink | Reply to this
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