The Stalks of a Presheaf

One more construction we’ll be interested in is finding the “stalk” of a presheaf over a point . 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 , 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 the collection of open neighborhoods of form a directed set, and we can take the limit .

Again, we’d like to understand this in more concrete terms, for when 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 is an element of for some neighborhood of . Two elements are considered equivalent if they agree on some common neighborhood of . That is, if we have and , and if there is some so that , then we consider and to be the same element of . 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 , 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.