Big to Small, Small to Big: Topological Properties through Sheaves (part 2)

Continuing from where we left off yesterday...

Yesterday, I managed to describe what a *presheaf* was. Today, I'm going to continue on that line, and get to what a full sheaf is.

A sheaf is a presheaf with two additional properties. The more interesting of those two properties is something called the *gluing axiom*. Remember when I was talking about manifolds, and described how you could describe manifolds by [*gluing*][glue] other manifolds together? The gluing axiom is the formal underpinnings of that gluing operation: it's the one that justifies *why* gluing manifolds together works.

Before the gluing axiom, there's another necessary property for a sheaf, called the *normalization axiom*. It's *much* easier to describe in terms of the second, category-theoretic definition of presheaf. If we have a presheaf *F* based on a category **C**, then for *F* to be a sheaf, the gluing axiom says that **C** must have a [terminal object][terminal] t, and *F*(∅) = t.

The gluing axiom for a sheaf is the really neat one. What it says is that we can glue
two subsets together if they agree on the overlap. To make that formal:

Suppose we have a topological space **T**, and a presheaf *F*.
Let { Ti }i=1..n be a collection
of open subsets of **T**; and since we'll want to look at the union of the open subsets in that set, let *U* be the union ∪i=1..n Ti.

For each open set Ti, we can use *F* to define a *section* si. Given those sections, we can say that two sections si and sj are *compatible* if/f ρTi∩Tj,Ti(si) = ρTi∩Tj,Tj(sj). (That is just a fancy way of saying that their restriction functions *agree* on the overlap.) We can extend the definition of *compatibility* to the entire set *U* by saying *U* is a compatible group of open subsets if all pairs of sets within it are compatible.

Now, finally, we can get to what the gluing axiom for a sheaf *F* says!

Given a topological space **T**, and a *sheaf* *F*, for all sets *U* of open subsets of **T** U={ui}i∈I with *compatible sections* { si }i ∈ I, there there exists exactly one section s ∈ F(U) such that &rhoui,U = si.

In simple terms, it says that if you use the restriction maps of compatible sections in a sheaf, they'll agree on *exactly one* mapping for the overlapped subsection. So the overlap is defined exactly once, and there can be no disagreement about the correct way of looking at it.

It's a lot of work to get to this point, but it's worth it. What we've done is go from the
informal but intuitive idea that *gluing* topological spaces together works, to a careful and precise definition of exactly what it *means* for it to work. And hopefully, if I've done an adequate job of explaining this, you can see why this says that gluing two manifolds results in a manifold.

You see, for something to be a manifold, what that means is that for all points in the space, there is a *local* property that the space around the point looks euclidean. But it's not enough for that *local* property to be true in just some places, it must be true *everywhere*.

By defining a sheaf over the topological space, we can easily show that that *local* property holds true for the entire space. And when we glue two manifolds together, we're forming a new topological space; the glue defines a mapping which we can use to define a sheaf for the entire new space; and if the glue is done right, the resulting sheaf *must* also show that the local property of being almost euclidean is true for the entire *new* space.

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Okay, I think I've gotten what you were trying to say, but it was confusing when you switched back from the all category-theory definition to the notation of the first one you gave yesterday.

I'd rewrite a paragraph above as: (unfortunately, the <sub> tag is disallowed in the comments)

For each open set Ti, call F(Ti) the section si. Given those sections, we can say that two sections si and sj are compatible if/f ÏTiâ©Tj,Ti(si) = ÏTiâ©Tj,Tj(sj). (That is just a fancy way of saying that their restriction functions agree on the overlap.) We can extend the definition of compatibility to the entire set U by saying U is a compatible group of open subsets if all pairs of sets within it are compatible.

Daniel:

No, F(T_i) is not a section. It is a set, and its elements are sections over T_i.

Okay, let's do this even more concretely (yes, category people can do concrete too). Consider the real line and four points aall of (a,d). We can do this because continuous functions form a sheaf.

By John Armstrong (not verified) on 13 Dec 2006 #permalink

then for F to be a sheaf, the gluing axiom says that C must have a terminal object

Should be "normalization axiom", I believe. Also, last \rho in gluing axiom lacks '#'.

Did you mean:

Consider the real line and four points a < b < c < d

Was there anything else in that paragraph that got dropped? (And, while we're at it, may we curse the programers who were unable to get the levels of escaping correct in Moveable Type's comment previewing system?)

I'll accept that what you said is what a section means, but if so then the original paragraph is in even more dire need of some serious cleanup, because that meaning of "section" is not apparent in the original. The original paragraph seems to imply that s_i is determined by F and T_i.

With this new meaning, the gluing axiom actually makes sense. I'm still not entirely sure how this relates to the gluing of manifolds; that is, how the introduction of sheaves does anything but royally screw up the very simple, intuitive notion of coordinate transformation functions. It's like bringing in a Rube Goldberg machine to hammer in a nail. It might, if you line up all the concepts just right, work for this purpose. However, a critical piece of this machine is the hammer anyway, only now you've got to go through a bunch of new creations to wield it.

Now, don't get me wrong - I'm all in favor of working at higher levels of abstraction when possible and am constantly annoying my coworkers with my generally functional style and invented higher-level functions within java. However, there's still something about this that smells like building castles in the air.

Also, in the discussion prior to the gluing axiom, Mark seemed to imply that the index set I was restricted to finite sets, and therefore that the axiom needs to hold only for finite collections of open sets.

I assume that in fact it must hold for arbitrary collections of open sets?

Yeah, I keep forgetting about html's nastiness with those symbols. Serves me right for learning TeX as a first markup language...

Okay, so how does this help us with manifolds? Well, I think Mark is getting to this. Really we need a notion of morphisms between sheaves in order to say it most elegantly. Until then, ponder on the sheaf of differentiable functions on a differentiable manifold.

By John Armstrong (not verified) on 14 Dec 2006 #permalink

Okay, I'll take your word for it that morphisms between sheafs make it work nicely.

Right now, it just seems like a bunch of unnecessary scaffolding going off to nowhere, when coordinate patches with differentiable coordinate transformations where patches overlap works so easily.

And there's nothing wrong with \TeX; in fact, I sometimes suspect that Mark writes most of his posts in \TeX and then translates it to html. There are often things from \TeX that I wish CSS had.

Daniel Martin: Coordinate patches? We don' need no steenkin' Coordinate patches. Really, how can those even exist if the space is not a Metric Space, or even semimetric space?

Sorry for the silence in response to the comments and questions, and more importantly for the sloppiness of the original post. After I finished part 1, I got news of a family
medical emergency; my father has been extremely ill, and had to go back to the hospital to have his leg amputated in an attempt to save his life from an infection. So I rushed through the second half just to get it posted before I had to leave to head for the hospital. I should have just let it wait. Anyway, I'll be reposting an improved, better version of this.

Mark, I'm sorry to hear that. Meanwhile, as for the first post on contravariant functors, the comments have cleared things up.