Sheafification of a presheaf

Question

Suppose we are given an arbitrary metric space $X$ and we want to construct a compact space from this. Equivalent condition for a metric space to be compact is that every sequence has a convergent subsequence. To construct a compact space from $X$ what we do is (at least in case of $\mathbb{R}$ that is what we have done) we consider all sequences and add limit points of these sequences to $X$. So, adding some points to $X$ and giving a topology fixes the possibility of having a sequence which has no convergent subsequence. This gives an injective continuous map $X\rightarrow \tilde{X}$ and we call this $\tilde{X}$ a compactification of $X$.

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Suppose we are given an arbitrary group $G$ and we want to construct an abelian group what we do is remove some elements (quotienting by commutator subrgroup $[G,G]$) from $G$ and we get an abealian group $\tilde{G}$ and a map $G\rightarrow \tilde{G}$ and we call this abelianization of $G$.

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So, the basic idea is if our set is too big to be something that we want, we remove something from it and if it is too small to be something that we want, we add something to it. I am trying to understand sheafification from this point of view.

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Suppose we are given a presheaf $\mathcal{F}$ and we want to construct a sheaf from this. To satisfy gluability condition, it is reasonable to think of adding some sections so that given an open cover $\{V_i\}$ of $U$ and sections $s_i\in \mathcal{F}(V_i)$ such that $s_i|_{V_i\cap V_j}=s_j|_{V_i\cap V_j}$ there exists $s\in \mathcal{F}(U)$ such that $s|_{V_i}=s_i$. I do not know how to think about identity axiom in this point of view. Any suggestion regarding this is welcome.

We define sheafification $\tilde{\mathcal{F}}$ of $\mathcal{F}$ as $\tilde{\mathcal{F}}(U)=\{s:U\rightarrow \bigsqcup \mathcal{F}_p \text{ satisfying some conditions }\}$. I want to know how do I see this as fixing the problem of Gluibility and Identity axiom. Any suggestions are welcome.

Answer 1

You need more fuel for analogy!

Suppose you have a set of vectors. This set is missing some linear combinations of elements! You want to solve the problem of filling in these missing linear combinations, while preserving whatever linear relations already exist between your vectors.

This is called the span of those vectors. We solve this problem in a very different way: these vectors were given to us as elements of some ambient vector space. We construct the span as a subset of that ambient vector space that satisfies the desired property.

The lesson here is that sometimes we already have some ambient notion that the thing we are trying to construct can be described in terms of.

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That is closer to what's going on with the definition of sheafification you reference. In nice cases (sheaves on topological spaces count as 'nice'), one of the things we know about the sheafification of a presheaf is that they have the same stalks.

So if we're given a presheaf $P$ and want to find its sheafification $S$, we already know the stalks of $S$. So we can assemble the stalks into a discrete bundle $E$, and then we know that $S$ is (isomorphic to) a subsheaf of the sheaf of sections of $E$. So all we need to do is identify which sections are the members of $S$!

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We can actually do better: we can give $E$ a topology that makes it locally homeomorphic to the base space, and it turns out that $S$ will be exactly the sheaf of sections of this bundle. We call this bundle the étale space of the sheaf.

(some sources even define a sheaf to be an étale space, since the category of sheaves on $X$ is equivalent to the category of spaces equipped with local homeomorphisms to $X$)

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Incidentally, describing things in terms of stalks solves the identity problem. Hartshorne's "some conditions" presumably can be clearly seen to solve the gluing problem (probably in terms of defining a section as something that can be covered by things). The étale space construction also solves the gluing problem, since sheaves of sections of bundles automatically have the gluing property.

Answer 2

Here are my reflections on the specific sheafification construction in Hartshorne. First, the paragraph in question:

We construct the sheaf $\overline{\mathcal F}$ as follows. For any open set $U$, let $\overline{\mathcal F}(U)$ be the set of functions $s$ to the union $\bigcup_{P\in U}\mathcal F_P$ of the stalks of $\mathcal F$ over points of $U$, such that

1. for each $P\in U$, $s(P)\in \mathcal F_P$, and
2. for each $P\in U$, there is a neighbourhood $V$ of $P$, contained in $U$, and an element $t\in \mathcal F(U)$, such that for all $Q\in V$, the germ $t_Q$ of $t$ at $Q$ is equal to $s(Q)$.

So, what is going on here? I like to think about the (ring-)sheaf $\mathcal F$ of analytical functions on $\Bbb C$ with the standard topology, because it's a very nice sheaf. Let's sheafify it! (Even though it's already a sheaf; we shall take a sub-presheaf afterwards and go through the same process.)

In order to sheafify, we need to know what the stalks are. In this case, the stalk $\mathcal F_P$ over a point $P$ is the ring of power series around $P$ with positive radius of convergence. For any element $s\in \mathcal F(U)$, the germ $s_P\in \mathcal F_P$ is the power series of $s$ expanded at $P$.

Let's first see what we get by only imposing 1. above, and not 2. This fives us a sheaf $\mathcal F'$ that consists of functions $s:U\to \bigcup_{P\in U} \mathcal F_P$ with only the restriction that $s(P)\in \mathcal F_P$. Where does that leave us? It means that a section of $\mathcal F'(U)$ consists of picking one power series at each point in $U$, with only the restriction that each power series should have positive radius of convergence. This is, in other words, a pretty large sheaf (it has a name: the Godement sheaf or Godement resolution of $\mathcal F$).

So, let's take this large sheaf $\mathcal F'$, and impose restriction 2, which gives us $\overline{\mathcal F}$. What does this restriction really say? It imposes some coherence restrictions on what power series we allow in a section. It says that given a section $s\in \overline{\mathcal F}(U)$, for each point $P$ there is a neighbourhood $V\subseteq U$ of $P$ and an analytic function $t$ on $V$ (this function is from $\mathcal F$, mind you!) such that for each point in $V$, the power series that $s$ defines at that point is the power series of $t$ at that point. In other words, in $\overline{\mathcal F}(U)$ we place the restriction that the power series given at each point by a section should all come from the same analytic function.

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Now, what happens if we take a sub-presheaf and work with that instead? This will show you how sections are added by sheafification, although sadly it won't show you how superfluous sections are killed.

Take the presheaf $\mathcal F$ of analytic functions on $\Bbb C$ with an antiderivative, and let $U = \Bbb C-\{0\}$. Thus, for instance, we have $\frac1z \notin \mathcal F(U)$. However, $\frac1z$ does have an antiderivative on $\Bbb C-(-\infty, 0]$ and on $\Bbb C-[0, \infty)$, which means that a section corresponding to $\frac1z$ could be made locally. This shows that $\mathcal F$ is not a sheaf (and incidentally, is the standard example of the fact that image presheafs of a map of sheafs isn't always a sheaf; $\mathcal F$ is the image presheaf of the differentiation map on the sheaf of analytical functions).

So, how does this section get added to our $\overline{\mathcal F}(U)$? The simple answer is that the function $\frac1z$, while it doesn't have an antiderivative on $U$, it does have a power series with positive radius of convergence on each point of $U$, and for each such point $P$, there is a neighbourhood $V\ni P$ such that $\frac1z$ has an antiderivative on $V$. That means that $\frac1z\in \mathcal F(V)$, and this plays the role of our $t$ above.

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As for how locally zero sections of presheafs get killed, they die once you pass to stalks. If $s\in \mathcal F(U)$ is non-zero, but locally zero, then for each point $P\in U$, the germ $s_P\in \mathcal F_P$ is zero. And since the image of $s$ over the canonical map $\mathcal F(U)\to \overline{\mathcal F}(U)$ is the function that to each point $P\in U$ assigns the germ $s_p\in \mathcal F_P$, this becomes the zero function.