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 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 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.
Comments
Did your html cause you to drop a chunk? What's a section? Where did ρ come from?
Posted by: Daniel Martin | December 13, 2006 7:39 PM
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)
Posted by: Daniel Martin | December 13, 2006 7:48 PM
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 a
We know what it means to have a continuous function on an interval, right? So let's consider functions f_1 on (a,c) and f_2 on (b,d). In the language of sheaves we say that f_1 is a "section" over (a,c) of the sheaf of continuous functions on R. The collection of all continuous functions on (a,c) is F((a,c)) in this example.
Since f_1 is continuous on (a,c), we can just forget about what happens outside (b,c). We say that f_1 restricts to a function g_1 on (b,c). Here's where a lot of people go wrong: f_1 and g_1 are different functions because they have different domains, but since they agree wherever they're both defined (on (b,c)) it's common to sloppily identify the two. Similarly, f_2 restricts to a function g_2 on (b,c). Again, g_2 is just the function that assigns to any point in (b,c) the value of f_2 at that point.
Now if g_1 and g_2 are the same function on (b,c) that means that given a point in (b,c) the functions f_1 and f_2 take the same value. Now we can glue them together to give one continuous function defined on all of (a,d). We can do this because continuous functions form a sheaf.
Posted by: John Armstrong | December 13, 2006 9:38 PM
Posted by: nikita | December 14, 2006 8:19 AM
Did you mean:
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.
Posted by: Daniel Martin | December 14, 2006 2:14 PM
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?
Posted by: Daniel Martin | December 14, 2006 2:20 PM
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.
Posted by: John Armstrong | December 14, 2006 7:56 PM
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.
Posted by: Daniel Martin | December 15, 2006 11:00 AM
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?
Posted by: Jonathan Vos Post | December 15, 2006 11:30 AM
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.
Posted by: Mark C. Chu-Carroll | December 15, 2006 2:01 PM
Mark, I'm sorry to hear that. Meanwhile, as for the first post on contravariant functors, the comments have cleared things up.
Posted by: Torbjörn Larsson | December 16, 2006 8:14 AM