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 $U_i$ to get a section on an open set that they cover, at least if these sections agree on the intersections $U_i \cap U_j$.
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 $X$, we think of it as a topological space and take its poset of open subsets, $O(X)$. Then we give this category $O(X)$ 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 $O(X)$ into a site — the little Zariski site of $X$.
To define the big Zariski site of a scheme $X$, we take the category of schemes, say $Sch$. Then we give this category a pretopology where a covering family is a jointly surjective family of scheme-theoretic open immersions. This makes $Sch$ into a site. The category of schemes over $X$, say $Sch/X$, then becomes a site in its own right — the big Zariski site of $X$.
(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 $X$ is called the petit topos while the sheaves on the big Zariski site of $X$ is called the gros topos.
Here are a bunch of questions:
- Do the petit and gros topos of $X$ fail to be equivalent as categories? Perhaps this already fails when $X$ 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 $Sch$ 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 $X$ 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!
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 $X$ (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.