I’ve mostly been taking it easy this week, since readership is way down during the holidays, and I’m stuck at home with my kids, who don’t generally give me a lot of time for sitting
and reading math books. But I think I’ve finally got time to get back to the stuff
I originally messed up about sheaves.
I’ll start by talking about the intuition behind the idea of sheaves. The basic idea of
a sheave is to provide a way of taking some local property of a topological space, and
demonstrating that it holds everywhere. The classic example of this is manifolds, where the *local* property of being locally almost euclidean around a point is expanded to being almost euclidean around *all* points.
To be able to do that step from the local to the universal requires a step of abstraction; it needs to exploit the particular structural properties of the relationships between points in the space. The easiest way to talk about the structural properties is by choosing a mathematical structure that’s appropriate for the property you want to study, and creating
a particular kind of correspondence between the open sets of the topological space (that is, the topological structure) and objects of the type that you want to use for reasoning.
To make that work, you basically need to do two different things: one is to generate the mapping between the open sets of the space in a way that preserves the important properties of the topological structure of the space; the second is to show that joining the mapped objects
preserves structure – that it, given objects corresponding to overlapping open sets, you can show that the topological properties are preserved when those objects are joined.
If you do the first step – showing a mapping between open sets and some other mathematical
objects in a structure-preserving manner – what you get is a *pre-sheaf*. If you take a
pre-sheaf, and show that it has the properties needed to making joining objects corresponding
to sets work, then you’ve got a sheaf.
So what’s a pre-sheaf? As I said above, a pre-sheaf *F* over a a topological space **T** is a mapping from open sets in **T** to some kind of objects where the mapping preserves the topological structure. Since the topological structure is defined by subset relations, that means that the mapping has to preserve the basic properties of the subset relations. Formally, you can described a pre-sheaf *F* as a mapping to a category *C* with a special set of morphisms, Ρ such that:
1. Every open set of **T** is mapped by *F* to an object in **C**.
2. For every pair of open sets x, y in **T** such that x ⊆ y, Ρ contains a ρx,y such that:
1. For all points *x* in **T**, ρx,x = the identity morphism for *F*(x). *(The mapping preserves the identity {*x*} ⊆ {*x*}.)*
2. For all triples of open sets M, N, and O in **T**, ρM,NºρN,O = ρM,O. *(The mapping preserves the transitivity of the subset relation: M ⊆ N and N ⊆ O ⇒ M ⊆ O.)*
Or alternatively, the presheaf can be seen as a *contravariant* functor from the topological category of **T** to the category **C**. The contravariant functor guarantees that the mappings will behave the way that the restriction morphism describe above.
Now, suppose we’ve got a presheaf *F* mapping to the category *C*; that’s called a *C*-valued presheaf. If we have an open set from **T** called *O*, then *F(O)* is a set which represents the part of the structure of **T** enclosed by the open set *O*. We call *F(O)* the *sections* of *F* over *O*. Assuming that the objects in *C* are sets (which is not *always* the case, but will be the case in pretty much every category we use to describe anything topological), then each element of *F(O)* is a *section* of *O*.
What we want to be able to do when we go from pre-sheaves to sheaves is to show that the way that the pre-sheaf represents the structure of **T** allows us to reason about properties of subsets like *O*, and then expand that reasoning to the entire topological space by showing that all subsets of *O* possess the desired property, *and* (this and is the really important part!) that when two subsets overlap, it’s *not* just the case that both subsets have the desired property, but that both subsets *agree* on how that property is represented in the sheaf on the sections where they overlap.
So how do we go about saying that?
Suppose we have a presheaf *F* over the topological space **T**. First, we need to show that the category *C* used by *F* has a necessary property for small-to-large reasoning to work. In category theoretic terms, *C* must have a terminal object *t*, and *F(∅)=t*. (A terminal object is sort of like a kind of universal lower bound; if the objects in a category are sets, the terminal object is the empty set.) This property is called the *normalization axiom*.
By far the more interesting part is the *gluing axiom* – that’s the part that lets us define
what it means for two subsets to agree on an overlapping section.
What does is mean for two subsets to *agree* on an overlapping section? We’ll start with a
slightly weaker notion, of *compatibility* between sections, and then from there, we’ll extend
it to *agreement* between sections. Suppose we have open sets *D* and *E*, and that
*D*∩*E*&neq;∅. Then if we do the presheaf mapping on *D* and *E*, we get a set of
sections over *D* and *E* in *F*. Two sections *sd* and *se* are
*compatible* if the restriction morphisms respect the rule *ρD∩E,D(sd) = ρD∩E,E(se)*. The open subsets *D* and *E* are compatible if/f for all pairs of sections *d∈F(D)* and *e∈F(E)*, *d* and *e* are compatible. (This is why the normalization axiom was important: if the sections do *not* overlap, there must be a unique value to represent that fact: the terminal object of the category.)
Now, finally, we can get to agreement, and what the gluing axiom requires for a presheaf to be a sheaf. Given a topological space *T* and a presheaf *F*, *F* is a sheaf if and only if for all sets *U* = {*ui*} of open sets of **T** with compatible sections *s={si}*, there is exactly *one* unique section **s**∈F(U) such that ρui,U(**s**)=si. The unique section **s** for
the set of open subsets *U* is called the *gluing* of the set *U*.
That’s a very fancy way of saying that whenever a group of open sets have overlapping sections, that the members agree that there is exactly *one* mapping between their respective sections, and that the structural properties in that section are consistent.
It’s worth reiterating here: the gluing axiom says *nothing* about coordinates, distances, or directions. All that it says is that the mapping of the open subsets of the topological space preserves their subset structure. So whatever properties the subset structure of the space has must also be represented in the sheaf mapping; and when sections with interesting properties
are glued together, the gluing respects those properties.
When we work with manifolds, we often do it using atlases, which are mappings from subspaces
of some euclidean space ℜN to sections of a manifold, and we glue together
sections of the those mappings. That works because of the gluing axiom, but the *coordinate
systems* that we get from doing that aren’t necessary for gluing; in fact, it’s the opposite: the metric structure of the sections is a local property, and the gluing axiom allows us to combine metrizable subspaces in a way that produces a single consistent metric for the entire manifold.
See, I think I’m with you, right up until the last paragraph where suddenly the logic swaps, and you’re suddenly saying the converse of what you’ve said up to then. It is as though you’ve got some hidden device which tells you a priori that atlases on manifolds form a sheaf. I would have said something like this:
Assuming, of course, that it is somehow useful to know that something is a sheaf beyond the direct fact of the gluing axiom, or that there is some method for showing that a bunch of atlases form a sheaf without doing the coordinate transformations in the first place. No one has yet told me whether these assumptions are true.
This is really my main objection – it appears that at some point either you or I swapped cause and effect. I really think that most of my confusion stems from this – you introduce the concept of sheaf and say that from this concept we get certain nice results about connecting global and local properties; I look at the results already on hand, see no way to showing that these structures are sheaves without going through all the local and global property results in the first place, and wonder why we’re going through sheaves to get to where we already are.
At this point, I see the concept of “sheaf” as nothing more than shorthand for “I’ve already proved separately that this particular property glues nicely.” Saying then that some particular property glues nicely because it is a sheaf makes as much sense as the classic Doctor Who bit about “dimensionally transcendental”:
“Why is the TARDIS bigger on the inside than the on the outside?”
“Because it’s dimensionally transcendental.”
“Oh. What does that mean?”
“That means it’s bigger on the inside than on the outside.”
At the risk of descending down the path of an inaccurate analogy, consider that the permutations of n objects form a group under composition of permutations. Now, they form a group because each permutation has an inverse, because there’s an identity permutation, and because the set of permutations is closed under composition. Once we know that they form a group, we can haul in some finite group theory to prove interesting statements such as “if p is a prime greater than n, then there is no permutation except the identity that can be repeated p times to get to the identity permutation”. However, it would be downright silly to say that we bring in group theory to show that each permutation of n elements has an inverse permutation. We need to show that to establish that the n-permutations form a group in the first place.
Martin:
The concept of the sheaf is fundamentally about two things:
(1) Defining what a *section* means for a topological space;
(2) Defining what it means for two open sets to *agree* on
a section.
You’ve got to remember that not all topological spaces are anything like what we think of as geometric spaces. When we’re talking about something like a manifold, it *seems* like it’s obvious how to define where things overlap, and what it means for them to agree. But that’s because the manifold is a space with a metrizable geometry. There are many topological spaces – like some of the spaces you encounter in domain theory (which is where I first encountered topology, by way of denotational semantics of programming languages) – which don’t correspond to simple geometries. In topological spaces like that, what it means for two open sets to overlap is *not* always clear. The sheaf gives you a tool that allows you to abstract the space into a form where you can precisely define overlap and agreement.
For a manifold, there is a sheaf of functions that defines the sections and satisfies the gluing axiom; and I will show you that sheaf in a post next week. But it seems obvious when you restrict yourself to manifolds that this *should* work; and you can prove the validity of gluing manifolds without resorting to sheafs. Sheafs come into their own when you’re looking at more complicated spaces – either higher dimensional ones where your ability to visualize falls apart, or spaces with no relation to a metrizable geometry.
Am I correct in supposing that the gluing axiom doesn’t make sense when the category C that our presheaf maps to is not a category whose objects are sets of some sort?
Does this mean that sheafs always map to a category made up of sets?
One thing that seems a little weird here is that you’re still saying that sheaves are primarily for studying the space itself, when (at least as I see it) they’re for studying structures on the space, like functions or fields or metrics. Patching together manifolds with the sheaf of atlases is a bit of hack. Patching smooth functions on the manifold is more to the point.
This seems semantically mismatched here. A sheaf defines what it means for two sections to agree on the intersection of their open sets.
Daniel:
(See John’s comments; they may help clarify thigns a bit.)
Sheafs do *not* always map to a category made up of sets. It’s convenient if they do, but the objects in the sheaf don’t have to be sets – they can be groups, vectors, vector bundles, functions, atlases… All manner of things with different kinds of structure.
I think that in my attempt to minimize the jargon level, I’ve reduced the clarity of some things rather than increasing it – which is where I think John’s comments really help. *Sections* are parts of a sheaf. They aren’t necessarily *sets*; they’re objects produced by the sheaf structure, and what they look like can depend on the nature of the sheaf.
As John said, what the gluing axiom does is define what it means for sections to agree about the intersections between their open sets. That is, sections are generated by sheaf mapping of an open set. The sections generated by the mappings from two different open sets have to agree about the parts of those open sets that overlap. The *sections* don’t overlap; the *sets* do. The *sections* define a kind of structure that is defined for the sets; the agreement means that both sections agree that the sets possess the structure of interest.
I’m confused. A group is a set. It’s a set with some additional structure, sure, but it’s still also a set. You can still say s∈F(U) if F(U) is a group and have that be meaningful without adding a whole new meaning to ∈. I guess what I’m asking is if there are any sheafs that map to categories that are not subcategories of Set. I’m a bit shaky on what such a category would be, but I am assured by others that such categories exist.
After posting that, I think I have an example.
Let S be {0,1}, a set of two elements, and give S the discrete topology. (i.e. all four subsets of S are open)
Now, consider that we have a category Top_S (silly comments won’t let me use superscript and subscript) that is formed by all open subsets of S with the only morphisms being the inclusion maps.
Let C be the dual category to Top_S. That is, the same objects but all the morphism arrows point the other way.
Now, F given by F(U) = U is then a pre-sheaf. Is it a sheaf? The normalization axiom holds, but I’m having a lot of trouble wrapping my head around what the gluing axiom means in this case.
Actually, there it’s better if you use a three-element set for S, since otherwise you don’t get interesting intersections. So let S be a three element set {0,1,2}, and the rest as before.
I’m surprised no one else caught this, but the terminal object in the category Set is any one-element set. (All of these are isomorphic)
The empty set is the initial object of Set, the dual concept.
Terminal is more like an upper bound in a partial order; the initial is the lower bound.
Marc, you’re right. Even better, it makes perfect sense.
Terminal objects in Set are singletons. F(∅) must be terminal. Therefore F(∅) must be a singleton (for F a sheaf of sets). Thus there is a unique section over the empty subspace.
Immediate consequence, all sections over disjoint subspaces are compatible, since their restriction to the (empty) intersection must be the unique section over the empty subspace. If it went the other way, then no sections over disjoint subspaces could be compatible. We couldn’t even really restrict to the empty subspace.