In this quick post, which is probably being written far too late at night to be remotely coherent, I want to lay out a few random thoughts I’ve been having about sheaves.

Classically, a sheaf is when you attach to a topological space some kind of compatible system of abelian groups (or sets, modules, whatever… I’ll stick to abelian groups to avoid having to think too hard). More specifically, each open set gets given a group , whenever there is some kind of relation , there has to be a restriction map , and there is a gluing condition that is supposed to guarantee good local behaviour. You can form, at each point , the *stalk* , and the gluing condition wins us the ability to check all sorts of things about sheaves by just looking at the stalks (whether a map is an isomorphism, injective, a sequence exact, etc.).

There are also (in all categories I’ve seen) an important pair of functors attached to every map . The most obvious is the *direct image* given by the definition (using the defining property of continuous maps)

There is also a functor going in the opposite direction, the *inverse image* given by taking the sheafification (basically where you modify the below as harmlessly as possible to force the gluing axioms to hold) of the presheaf defined by

These are *adjoint functors* in the sense that to give a sheaf map is equivalent to giving a map in a natural way. It turns out that is actually an exact functor (this is reasonably obvious looking at the definition and noting that direct limits and sheafification are exact operations). It follows entirely formally that is left exact and preserves injectives. There is also usually another functor which acts as a *left* adjoint to . These functors are basically really nice, but also manifest themselves in lots of different ways.

For example, consider the inclusion of an open subset. Then is just the obvious restriction of to . Another interesting example is if you just consider the inclusion of a single point . Looking at the definitions, it is clear that one gets none other than the *stalk* . In other words, by considering very special cases we can recover all the basic sheaf operations from the inverse image functor. This has two advantages. Firstly, it helps us remember that they are exact, and have right and left adjoints. Secondly, and more significantly, it seems to suggest an easier way to think about stalks in more general situations than the classical case.

It is a reasonably trivial observation that a presheaf (sheaf minus gluing axioms) is a contravariant functor from the category of open subsets of our space (with arrows given by inclusions) to the category of abelian groups. One could therefore contemplate calling any contravariant functor a *presheaf*.

But here is an interesting idea: let’s take to be the category of *all* topological spaces, but throw away all the arrows which don’t correspond to inclusions of open sets. We can still define a presheaf on this much larger category. However, with a little thought it’s clear we can also define a *sheaf*. The gluing condition passes more or less unchanged. For every open set and a cover of it , a bunch of compatible sections on the (note that it’s important that will be an open subset of and so the category comes equipped with the relevent arrow) must glue to a unique section on . The only difference is that now the open sets keep on going upwards and upwards forever through the celestial heights of the category of topological spaces. And hey, why not? This is kinda fun!

This example feels slightly strange. Why? It’s no longer functions on a fixed space. Somehow, things have become sort of unbounded. But is that really what’s strange? I think what is actually strange about this example is the artificiality of the category of topological spaces where the only maps are open embeddings. This was what Grothendieck thought too, and pretty soon he was able to define sheaves on arbitrary categories provided they came equiped with a gadget called a Grothendieck topology, whose job is to track what the `covers’ are for which the gluing conditions need to be checked.

Let me be (sort of) precise again. Let be the category on which we want to construct sheaves, and assume it has binary pullbacks (just as we needed pairwise intersections of open sets above). A Grothendieck topology on it is just, for each object , a collection of covers . A sheaf is then just a contravariant functor satisfying a gluing condition exactly analogous with the classical one. In these more general situations, certainly whenever I’ve come across them, there are also good generalisations of the functors I mentioned earlier and they retain their properties.

The idea of sheaves is one of the most powerful in 20th century topology and geometry, and what we have outlined is a vast generalisation of this concept, so perhaps it is unsurprising it has been hugely successful. However, a reasonably humble question has been bothering me for a while. Since there is no underlying space anymore, there are no longer really such a thing as points, so we cannot obviously define stalks (and check properties locally at them). This seems to be a shortcoming that we should try to overcome.

We have already seen part of the way to an answer. Does our category contain any sort of object that deserves to be called a point? If so, let us call the *points* of an object the set of maps . Then given a sheaf on our category (or at least on an appropriate subcategory of things mapping into ), we can finally define the stalk .

But wait? We want our stalk to be an actual abelian group, whereas we’ve got ourselves a whole sheaf here. Therefore, having boldly gone forth and defined sheaves to be on entire large categories, I now need to retrace my steps to avoid losing grip on reality. Given a sheaf on , and an object , we may restrict our attention to the subcategory of given by taking covers of , and covers of all the objects involved therein, and so on… Ignoring any logical/set-theoretical problems this might entail, we write for the category of sheaves on this subcategory. Our functors will then give to any map appropriate maps between categories of sheaves.

So what should look like? Well, we just want to be as trivial a category as possible. Indeed, if it were a one point category, it would just be an abelian group, so the stalk is an abelian group as required.

Generally I have no idea how possible this is, but let’s have a look at the category of schemes with any sensible topology. What are the most likely candidates to be points? Well, probably schemes which have a single point (as topological spaces), that is, fields . Indeed, maps do correspond reasonably nicely to actual points of . However, in many topologies, for example the etale topology, a sheaf on is not simply an abelian group but rather a -module. It therefore seems that the most sensible candidates for points are the schemes for a separably closed field: and these are indeed what we take as the `geometric points’ and to form `geometric fibres’ in scheme theory.

This also gives a quick generalisation of the concept of skyscraper sheaf. Indeed, given a geometric point , while gives the correct definition of stalk, one can also form the direct image of any abelian group and it gives a sheaf , called the skyscraper sheaf, whose classical analogue is characterised by

I find it amusing just how nice so many of these sheaf-theoretic constructions seem to be, and how often trying trivial things can shed new light on things and provide the correct generalisations of classical ideas. Anyway, it’s far too late, and I hope I didn’t ramble too much.

## 2 comments

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March 13, 2012 at 11:10 am

Zhen LinI’m glad to see you are picking up topos theory. :p The simplest object in the category of $k$-schemes is $\operatorname{Spec} k$ itself: it has no automorphisms whatsoever. Unfortunately this leads to a poor notion of point, unless $k$ is algebraically closed.

You could define an abstract point of a abelian sheaf category $\mathcal{A}$ as follows: it is any pair of adjoint functors $x^* \dashv x_* : \textbf{Ab} \to \mathcal{A}$ such that the left adjoint is left exact. I’m not sure what this recovers in general; but if you’re willing to work with _toposes_, meaning set-valued sheaves, then it is known that the set of abstract points of $\textbf{Sh}(X)$, where $X$ is a sober topological space, naturally has a topology and is homeomorphic to $X$. In particular, if $X$ is the underlying space of a scheme, $X$ is sober, so this construction is seen to be the correct one in this case. (In general, if $X$ does not have good separation properties, $\textbf{Sh}(X)$ may have too many or too few points.)

I think the existence of $f_!$ is something very special, even in the case of topological spaces. I think it’s the “extension by zero” functor, when you’re working with abelian sheaves.

March 13, 2012 at 7:51 pm

tloveringThanks for the reply. My knowledge of general topos theory is pretty hazy (mainly the few vague things I picked up from Caramello’s course last year, and the things that are useful for understanding the etale site), so it’s handy to get an idea of how much the elementary sheaf theory generalises to arbitrary Grothendieck toposes.