The topology posts have been extremely abstract lately, and from some of the questions
I’ve received, I think it’s a good idea to take a moment and step back, to recall just
what we’re talking about. In particular, I keep saying “a topological space is just a set
with some structure” in one form or another, but I don’t think I’ve adequately maintained
the *intuition* of what that means. The goal of today’s post is to try to bring back
at least some of the intuition.
So let’s recall just what a topological space is. Our definition from the [very beginning of
the topology series was:][top-space]
A topological space is a set **X**, and a collection **T** of subsets of **X** where the following conditions hold:
1. ∅ ∈ **T** and **X** ∈ **T** *(The empty set and the entire set **X** are both members of **T**.)
2. (∀ C ∈ **2****T**): (∀ c ∈ C : ∪(c ∈ C)c ∈ **T**) *(The union of any set of elements of **T** must be a set in **T**.)*
3. ∀ s,t ∈ **T**: s ∩ t ∈ **T** *(The intersection of any pair of members of **T** is also a member of **T**.)*
**T** is the structure imposed on the set **X** that we’ve been talking about. But just what does that mean? It’s really a very fancy way of taking the concept of *closeness* or *adjacency* and abstracting it out so the concept of *distance* isn’t needed. In a topological space, we don’t care whether we can measure *how far* it is from a point *A* to a point *B*; but we *do* care
whether we can meaningfully ask “Is B closer to A than C?” or “Is A adjacent to B?”. The
structure of the open subsets in a topological space gives us a way of talking about that.
How can we answer those questions? By playing with neighborhoods – that is, expanding “shells” of points around a particular given point. Suppose we want to ask “**Is *B* closer to *A* than it is to *C*?**”
Take a sequence *S* of expanding subsets around *B* something like the open balls in a metric space – that is, a sequence of subsets that are uniformly growing larger, but always including all of the points in the subsets that precede them. If *A* becomes an element of the sets in the sequence *before* *C* does, then *with respect to* the sequence *S*, *A* is closer to *B* than *C* is. In a topological space, you *cannot* in general define something like *closer to* in a universal way; there are many ways that the open sets can be constructed, and it’s entirely possible to have *many different* ways of describing closeness based on different
constructions, and there’s no reason to prefer one of them over the other.
That basic concept – what points are *next to* what points, and what points are *close to* what other points – is what’s defined by the open-set structure of the topological space. The notion of “close to” is based completely on subset inclusion relationships; you can’t necessarily assign a number to the distance between two points (if you could, we’d call it a metric space!), but you can always look at the subset inclusion relationships to understand where the points lie in relation to each other.
To bring this forward a bit, in my messed up post about sheaves, one of the key ideas was
the gluing axiom. The gluing axiom says, very basically, that you can map between *sets* in an overlap between two sections; it does *not* do a coordinate transformation. That misunderstanding was caused by a combination of some genuinely subtle distinctions, and some
dreadful sloppiness on my part.
When we talked about gluing manifolds, what we were doing is forming manifolds by mapping
sections of *euclidean spaces* onto manifolds, and gluing them together with coordinate
transformations. That *is* gluing, and the theoretical basis for it *is* sheaves and
the gluing axiom that allows them to be combined. But the important distinction is that what we were gluing was sections of euclidean spaces – and euclidean spaces have a standard metric, and we describe the glue maps in terms of that standard metric.
[top-space]: http://scienceblogs.com/goodmath/2006/08/topological_spaces.php
Please define the word “section” as you use it in the last paragraph. I’m still having trouble with that word, as it seems to not at all be related to the way that “section” was used in the sheaves post.
Okay, on a re-read it seems that you’re using section there in the conventional sense of the word.
I still don’t understand though why it appears that you want to divorce yourself from coordinates. If you’re discussing manifolds, you’re considering objects which are everywhere locally euclidean. This means that you can cover the manifold with a bunch of coordinate patches. Sure, if we’re not in a metric space those coordinates may not mean anything “really” – I don’t claim that the coordinates impose any kind of well-defined metric on the space.
But now, once we have all those coordinates and coordinate transforms, there’s this other structure introduced that brings in all the machinery of category theory to make it all generic. Except that once again, in the case of gluing things together that are locally euclidean, this machinery tells you that gluing involves…
Coordinate transformations.
I reference my “Rube Goldberg” comment from the last sheaves post.
So is the point of sheaves to extend the gluing concept to things which are not locally euclidean everywhere?
Daniel:
We *aren’t* limiting ourselves to just manifolds – we’re talking about topological spaces *in general*. I’m going to write more about the gluing axiom, but for now, the short version is that the gluing axiom isn’t just used for building manifolds by gluing.
Sheaves and gluing are *not* just for splicing together things that are locally euclidean everywhere. The point of sheaves is that they allow you to take a local property – like “the space appears euclidean around a point p”, and expand from that local property to a global property: “for all points p, the space around p appears to be euclidean”.
Firstly, Mark is right that the notion of a sheaf is far more general than something we can do over manifolds. There’s a bit more to it than that. For one thing, sheaves don’t only apply to the manifold construction.
A sheaf of foos (where foo can be all sorts of different kinds of things, like sets, groups, rings…) is a much more general concept. For manifolds, I think what we’re looking at is a sheaf of sets where the set above an open subspace is the collection of atlases for that subspace, up to compatibility. I forget if this was mentioned before, but an atlas for a manifold is a collection of coordinate charts covering the manifold. I want to consider two atlases on a subspace to be “the same” if they are compatible.
Okay, so if we’ve got a manifold and an atlas, we can restrict the charts to any submanifold to get an atlas on that submanifold. In fact, if we restrict further to a smaller subspace we get the same atlas as if we just restricted directly from the whole space to the smaller subspace. That makes this construction into a presheaf — for every open subspace we have a set (the atlases for that subspace) and if one subspace sits inside another we have a restriction map.
Now if we have two overlapping subspaces and an atlas on each, we can restrict each atlas to the intersection. If the two restricted atlases are compatible, we can throw all the charts from both subspaces together to get an atlas for their union. This means that our presheaf is actually a sheaf.
We view this as a construction — “gluing manifolds” — by taking two manifolds and considering them to sit inside some larger topological space, overlapping a bit. We check that the differentiable structures are compatible on that overlap, and then we get an atlas for the larger space. That makes it a manifold too.
Sheaves are also tremendously useful for other things than such constructions. For example, given any fiber bundle over a manifold we can restrict the bundle to a subspace and consider the set of sections of the restricted bundle (this is where “section” in sheaf theory comes from). If the fibers carry additional structure, then the sets of sections will as well, and we get more interesting sheaves.
For example, every manifold has a tangent bundle, which is a fiber bundle with vector spaces as the fibers. This means that the collection of vector fields (sections of the tangent bundle) over some subspace is actually a vector space, and we get a sheaf of vector spaces. Functions on a subspace form a ring, so we get a sheaf of rings from them. In fact, since we can multiply a vector field over a given subspace by a function on that subspace we get a “sheaf of modules over a sheaf of rings”, and these structures behave in Sh(M) (the category of sheaves on M) very much like regular modules and rings behave in Set.
And if all that is too high-minded, consider the simple case of the sheaf of holomorphic functions on the complex plane. Every function has its domain of definition, we can restrict to smaller domains, and we can patch functions together into larger domains as long as they agree on the overlap of their domains. This is the sort of thing that often gets blurred in the lower-level discussions because we tacitly assume a function to have the largest domain where its expression makes sense.
But now say I want to talk about the Gamma function. I can’t give you a closed formula for it, but I can pick a point and expand a Taylor series for it. Unfortunately, the series only converges within a circle of some finite radius, and a lot of the domain of the function is left outside. How do I fix that? I pick another point and expand another Taylor series. Each Taylor series defines a holomorphic function in its disk, and where the disks overlap the functions agree. So I can patch the two of them together to make a holomorphic function over a larger domain. I keep doing this until I’ve got all the points in the plane covered but those where Gamma has a pole. Why can I take all these Taylor series and patch them together into one big function? Because each is a section over its domain in the sheaf of holomorphic functions on the plane, and being able to patch is what makes this a sheaf.
I’m certain that this explains completely where the term section comes from, but having never heard (*) of what fiber bundles are, I find this rather unenlightening. I’ll go spend some time wandering around MathWorld to see if it does any good.
But back to your example I do understand, of the Gamma function being patched together: you seem to be saying that the reason that you can patch together disks of holomorphic functions into a holomorphic function because holomorphic functions on C form a sheaf.
But isn’t the reason that holomorphic functions form a sheaf that patching like this works? Don’t you need to prove that you can do patching like this before you can say that holomorphic functions form a sheaf? Are there other ways to prove something a sheaf?
Is there a decent example of something that is a presheaf, and on which everything holds for it to be a sheaf except the gluing axiom? That is, some example of compatible sections that nevertheless fails to define a compatible section on the union of the sections’ domains, or fails to define one uniquely?
(*) for “heard” I suppose I should probably say “understood”. My graduate algebra teacher went on for weeks on an excursion into algebraic topology that no one in the 4-person class followed at all. (He was exceedingly bad at reading the body language of “smile and nod” as “I haven’t understood a single thing you said in three days”) I vaguely remember terms such as “fiber”, “bundle”, “sheaf” and “scheme”, but having at that point no category theory background let alone even an inkling of what kind of structures algebraic topology is trying to study, not much stuck.
Okay, a fiber bundle is basically a twisted up version of a product space. Let’s take this in steps.
A trivial fiber bundle E over a base space B with fiber F is the product space FxB. This has a projection map to B. Basically, above each point of B is a copy of F (the fiber over the point), and the product topology puts all the fibers together into the space E.
A fiber bundle in general is a “total space” E, a “base space” B, and a projection map p from E onto B so that for every point x the preimage p^{-1}(x) of the point is homeomorphic to some space F. Given a subspace A of B we can consider the preimage p^{-1}(A) of the subspace as a subspace D of E, and projection restricts to a projection p|D from D to A. That makes (D,A,p|D) a “subbundle” of (E,B,p). We insist that B have an open cover by sets {U_i} so that the subbundle over each U_i is trivial.
Examples: Let B be the circle and F be the interval. The product space is the cylinder. On the other hand, we can cover the circle by two arcs and glue the product (square) over one arc to the product (square) over the other, but flip the interval over on one of the two overlaps. This is the Möbius strip. Both the cylinder and the Möbius strip are fiber bundles over the circle with the interval as the fiber, but they are topologically distinct.
So given a bundle we can consider continuous maps f from B to E which sends each point to a point in its fiber. That is, p(f(x)) = x for all x. Such an f is called a section of the bundle.
Now if we have a bundle and a subspace U of the base space we can restrict the bundle to U and consider the set of sections of the restricted bundle. Call this set Gamma(U). If we have a smaller subspace V, then we can restrict sections over U to sections over V, which gives a map from Gamma(U) to Gamma(V). It turns out that this gives a sheaf.
Hey, can you slow down a bit in the comments? You’re going to leave me nothing to write about! 🙂