Franklin's Notes

Sheaf of germs

Suppose we have a topological space with underlying set $I$ and open sets given by $\Theta$. Given any point $i\in I$, we may define an equivalence relation $\sim_i$ on the open sets in $\Theta$ as follows: if $U,V\in\Theta$, we will say that $U\sim_i V$ iff there exists an open set $W$ containing $i$ such that $W\cap U = W\cap V$. In other words, $U$ and $V$ "look the same" if we get "sufficiently close" to $i$. We might visualize this situation as follows:

Note how in this picture, $U$ and $V$ are very different open sets, but if we restrict our view to the open set $W$, they look the same, so that $U\sim_i V$. The equivalence class $[U]_i$ of all open sets of $(I,\Theta)$ that are equivalent to $U$ under $\sim_i$ is called the germ of $U$ at $i$. Intuitively, $[U]_i$ encodes information about "the points of $U$ that are arbitrarily close to $i$", although this is somewhat misleading because there may very well be no points of $U$ that are "arbitrarily close" to $i$, for instance if $i$ is Hausdorff. Another way of interpreting the germ $[U]_i$ is as a way of measuring the "degree of containment of $i$ in $U$". Rather than just discriminating between points in $U$ and points not in $U$, we can discriminate between various different degrees of containment, with "fully contained" corresponding to the germ $[I]_i$ and "separated" corresponding to the germ $[\varnothing]_i$ - but there are many different germs in between!

The stalk over $i$, denoted $\Omega_i$, is defined as and it can be conceptualized as "the set of all possible local behaviors at $i$". We define $\hat I$ to be the union of all of the stalks: and we may define a topology $\hat\Theta$ on $\hat I$ by letting the basis sets take the form Finally, we can define a sheaf $\Omega:\hat I\to I$ as the first projection map sending $\langle i,[U]_i\rangle\mapsto i$, since this map is open (it preserves the basis sets).

The following is an example that I find particularly illustrative. Consider the topology on the set $\omega+1$ with basis elements given by the singleton sets ${n}$ and the intervals ${x ~ : ~ x \geq n}$ for the finite numbers $n\in\omega$. We can picture the basis elements as follows:

The open sets of this topology can take one of two forms: they can either be arbitrary sets of natural numbers, or they can be sets of natural numbers containing all naturals past a certain point as well as $\omega$. For any $n\in\omega$, there are only two possible germs at $n$, namely $[\varnothing]{n}$ and $[X] _n$, since each of these points is open, meaning that the only possible "local behaviors" at $n$ are to either contain $n$ or not contain $n$. However, there are many different germs at $\omega$, namely all of the possible different "end behaviors" of subsets of natural numbers, so that two subsets are considered equivalent if they either both contain or both omit all natural numbers past a certain point. There is also one additional germ, namely $[X]\omega$, the only possible germ of open sets containing $\omega$.




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