The Unapologetic Mathematician

Mathematics for the interested outsider

Sheaves

For the moment we will be more concerned with presheaves, but we may as well go ahead and define sheaves. These embody the way that not only can we restrict functions to localize them to smaller regions, but we can “glue together” local functions on small domains to define functions on larger domains. This time, let’s start with the fancy category-theoretic definition.

For any open cover \{U_i\}_{i\in I} of an open set U, we can set up the following diagram:

\displaystyle\mathcal{F}(U)\to\prod\limits_{i\in I}\mathcal{F}(U_i)\rightrightarrows\prod\limits_{(i,j)\in I\times I}\mathcal{F}(U_i\cap U_j)

Let’s talk about this as if we’re dealing with a sheaf of sets, to make more sense of it. Usually our sheaves will be of sets with extra structure, anyway. The first arrow on the left just takes an element of \mathcal{F}(U), restricts it to each of the U_i, and takes the product of all these restrictions. The upper arrow on the right takes an element of \mathcal{F}(U_i) and restricts it to each intersection U_i\cap U_j. Doing this for each U_i we get a map from the product over i\in I to the product over all pairs (i,j). The lower arrow is similar, but it takes an element in \mathcal{F}(U_j) and restricts it to each intersection U_i\cap U_j. This may look the same, but the difference in whether the original set was the first or the second in the intersection makes a difference, as we shall see.

Now we say that a presheaf \mathcal{F} is a sheaf if and only if this diagram is an equalizer for every open cover U_i. For it to be an equalizer, first the arrow on the left must be a monomorphism. In terms of sets, this means that if we take two elements s\in\mathcal{F}(U) and t\in\mathcal{F}(U) so that s\vert_{U_i}=t\vert_{U_i} for all $latex U_i, then s=t. That is, elements over U are uniquely determined by their restrictions to any open cover.

The other side of the equalizer condition is that the image of the arrow on the left consists of exactly those products in the middle for which the two arrows on the right give the same answer. More explicitly, let’s say we have an s_i\in\mathcal{F}(U_i) for each U_i, and let’s further assume that these elements agree on their restrictions. That is, we ask that s_i\vert_{U_i\cap U_j}=s_j\vert_{U_i\cap U_j}. If this is true for all pairs (i,j), then the product \left(s_i\right)_{i\in I} takes the same value under either arrow on the right. Thus it must be in the image of the arrow on the left — there must be some s\in\mathcal{F}(U) so that s\vert_{U_i}=s_i. In other words, as long as the local elements s_i\in\mathcal{F}(U_i) “agree” where their domains overlap, we can “glue them together” to give an element s\in\mathcal{F}(U).

Again, the example to keep in mind is that of continuous real-valued functions. If we have a continuous function f_U:U\to\mathbb{R} and another continuous function f_V:V\to\mathbb{R}, and if f_U(x)=f_V(x) for all x\in U\cap V, then we can define f:U\cup V\to\mathbb{R} by “gluing” these functions together over their common overlap: f(x)=f_U(x) if x\in U, f(x)=f_V(x) if x\in V, and it doesn’t matter which we choose when x\in U\cap V because both functions give the same value there.

So, a sheaf is a presheaf where we can glue together elements over small domains so long as they agree when restricted to their intersections, and where this process defines a unique element over the larger, “glued-together” domain.

March 17, 2011 Posted by | Topology | 12 Comments

   

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