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.

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March 22, 2011 - Posted by | Topology

3 Comments »

  1. You’ve got the wrong limit there in the third paragraph. You need the arrow pointing to the right to make this the direct limit, not the inverse limit.

    Comment by Greg Friedman | March 23, 2011 | Reply

  2. fixed

    Comment by John Armstrong | March 23, 2011 | Reply

  3. [...] of Functions Let’s take the structure sheaves we defined last time and consider the stalks at a point . It turns out that since we’re working with sheaves of -algebras, we can sort of [...]

    Pingback by Germs of Functions « The Unapologetic Mathematician | March 24, 2011 | Reply


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