The Unapologetic Mathematician

Mathematics for the interested outsider

The Stalks of a Presheaf

One more construction we’ll be interested in is finding the “stalk” \mathcal{F}_x of a presheaf \mathcal{F} over a point x. We want to talk about how a presheaf behaves at a single point, but a single point is almost never an open set, so we need to be a bit creative.

The other thing to be careful is that we’re actually not concerned about behavior at a single point. Indeed, considering the sheaf of continuous functions on a space X, we see that at any one point the function is just a real number. What’s interesting is how the function behaves in an infinitesimal neighborhood around the point.

The answer is to use the categorical definition of a limit. Given a point x\in X the collection \mathrm{Subset}(X)_x of open neighborhoods of x form a directed set, and we can take the limit \varinjlim_{\mathrm{Subset}(X)_x}\mathcal{F}.

Again, we’d like to understand this in more concrete terms, for when \mathcal{F} is a set, or a set with some algebraic structure attatched. It turns out that if we unpack all the category theory — basically using the existence theorem — it’s not really that bad.

An element of the stalk \mathcal{F}_x is an element of \mathcal{F}(U) for some neighborhood U of x. Two elements are considered equivalent if they agree on some common neighborhood of x. That is, if we have f\in\mathcal{F}(U) and g\in\mathcal{F}(V), and if there is some W\subseteq U\cap V so that f\vert_W=g\vert W, then we consider f and g to be the same element of \mathcal{F}_x. They don’t have to be the same everywhere, but so long as they become the same when restricted to a sufficiently small neighborhood of x, they’re effectively the same.

Our usual category-theoretical juggling can now reassure us that the stalks of a sheaf of groups are groups, the stalks of a sheaf of rings are rings, and so on, all using this same set-theoretic definition.

March 22, 2011 Posted by | Topology | 3 Comments