# The Unapologetic Mathematician

## 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