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.
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March 22, 2011 -
Posted by John Armstrong |
Topology
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.
fixed
[...] 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 [...]
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