The n-Category Café

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 $X$ (I’ll stick to the English terms) is that they’re both worlds of spaces over $X$, though in the small topos, the spaces are so similar to $X$ that you almost might as well be talking about $X$. 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 $k[x]/(x^2)$ but they are all the identity on the small topos.

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

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

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 $X$ we can consider sheaves on its ordinary site of open sets, givin a petit topos. Or we can consider sheaves on $Top/X$, giving a gros topos.

Somewhere in between, we can consider sheaves on $LH/X$, where $LH$ is the category of topological spaces and local homeomorphisms. It turns out that $LH/X$ is actually equivalent to the petit topos of $X$, 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 $Sch/X$ and $Top/X$ 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.

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.

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.

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

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

The data of the sheaf of rings $O_X$ thus gives you the relation of $X$ with all the schemes of finite presentation over $Spec(\mathbf{Z})$. As any ring is a filtered colimit of rings of finite presentation, you can understand from this all the sets $Hom(X,Y)$ for any scheme $Y$, hence recover $X$ 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 $X$ from the category $Sch/X$ and from the petits topoi of schemes (locally) of finite presentation over $X$. Indeed, we have a (pseudo-)functor $$(-)_{Zar}: Sch/X\to Topoi$$ which associates to a scheme $Y$ over $X$ its petit topos $Y_{Zar}$. Then the gros topos of $X$ can be described as the lax $2$-colimit of this pseudo-functor. In particular, the canonical morphisms of topoi from $Y_{Zar}$ to the gros Zariski topos of $X$ form the universal ‘lax cocone’.

This is a general process which works for some other different Grothendieck topologies on $Sch/X$. For instance, one might be interested by henselian local rings (a commutative ring with unit $R$ is henselian if, any unital polynomial with coefficients in $R$ which has a root in the residue field of $R$ has a root in $R$). The Nisnevich gros topos of $Spec(\mathbf{Z})$ (i.e. the topos of Nisnevich sheaves on $Sch/X$) is the classifying topos for henselian local rings. 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 $X$. 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 ‘$\alpha$-small presentation’ for some given cardinal $\alpha$, which might even look like no size condition at all if $\alpha$ is the cardinal of some universe; this corresponds then to the très très gros topos).

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