Sheafification of a presheaf

Definition:  Let \( \mathbf{F}\) and \( X \) be topological space, and \( \pi : \mathbf{F} \rightarrow X \) be a surjective map, then the triple  \( ( \mathbf{F}, \pi, X ) \) is called an ètale space if \( \pi \) is a local homeomorphism, i.e. for all point \( p \in \mathbf{F} \), there exists an open neighborhood \( W \) of \( p \) such that \( \pi(W) \) is open in \( X \) and \( \pi|_W : W \rightarrow \pi(W) \) is a homeomorphism. 

\( \Omega \) will be used to denote the topology of \( X \) and \( \Omega_x \) for all the open subsets of \( X \) containing the point \( x \in X \).

For all \( x \in X \), we say that \( \mathbf{F}_x := \pi^{-1}(x) \) is a stalk of \( \mathbf{F} \) at \( x \). If \( U \in \Omega \), \( \Gamma(U, \mathbf{F}) \) will denote the set of continuous maps  \( \alpha: U \rightarrow \mathbf{F} \) such that \( \pi \circ \alpha = id \). The elements of \( \Gamma(U, \mathbf{F}) \) will be called a section of \( \mathbf{F} \) over \( U \).

Let \( \mathbf{F}(U) = \Gamma(U, \mathbf{F}) \) for \( U \in \Omega \). Because the section is just continuous functions, there exists a natural restriction map. Namely, if \( V \subset U \) and \( \alpha \in \mathbf{F}(U) \), then \( res_{U,V}(\alpha) = \alpha|_V \), the function restriction. This turns \( \mathbf{F} \) into a presheaf of sets.

Because sections are simply maps, we can easily check the sheaf axioms. Let \( \{ U_i \} \) be an open cover of \( U \in \Omega \). If \( \alpha_1, \alpha_2 \) agree on the covers, then 

\[ \alpha_1(x) = \alpha_1|_{U_i}(x) = \alpha_2|_{U_i}(x) = \alpha_2 \]

 as \( x \in U \), so \( x \in U_i \) for some \( i \). 

Also, if \( \alpha_i \in \mathbf{F}(U_i) \), then we can define \( \alpha(x) = \alpha_i(x) \) where \( U_i \) is an open cover containing \( x \). If \( U_j \) is another cover that contains \( x \), then we have that it agrees on the intersection, so the above definition is well-defined. And by our construction \( \alpha|_{U_i} = \alpha_i \) is clear.

Hence \( \mathbf{F} \) is, in fact, a sheaf. To distinguish \( \mathbf{F} \) which is a topological space, we will denote by \( \mathbf{F}^b \) for the sheaf associated with the ètale space \( \mathbf{F} \). 

Proposition: Let \( ( \mathbf{F}, \pi, X) \) be an ètale space, then there exists a natural bijection between the stalks \( \mathbf{F}_x \) and  \( \mathbf{F}^b_x \). 

Proof) Given \( \alpha \in \Gamma(U, \mathbf{F}) \) with \( U \in \Omega_x \), we define \( j_U : \Gamma(U, \mathbf{F}) \rightarrow \mathbf{F}_x \) defined by \( \alpha \mapsto \alpha(x) \).  We have \( \pi \circ \alpha = id \Rightarrow \pi(\alpha(x)) = x \Rightarrow \alpha(x) \in \pi^{-1}(x) = \mathbf{F}_x \).

Now let \( \alpha_x \in \mathbf{F}^b_x \), then there exists \( \alpha \in \Gamma(U, \mathbf{F}) \) representing \( \alpha_x \). We define a map

\[ j_x : \mathbf{F}_x^b \rightarrow \mathbf{F}_x \]

defined by \( j_x(\alpha_x) = \alpha(x) \). Given two \( \alpha_1 \) and \( \alpha_2 \) representing \( \alpha_x \), call the open set that the two maps agree on by \( U \), i.e. 

\[ \alpha_1|_U = \alpha_2|_U \]

As \( U \) must contain \( x \), we see that \( \alpha_1(x) = \alpha_1|_U(x) = \alpha_2|_U(x) = \alpha_2(x) \). Hence \( j_x \) is a well-defined map.

Surjectivity: Let \( p \in \mathbf{F}_x \). Then \( p \in \mathbf{F} \). There exists a open neighborhood \( W \ni p \) such that \( \pi|_W : W \rightarrow \pi(W) \) is a homeomorphism. Let \( \alpha = (\pi|_W)^{-1} \). We claim that \( \alpha_x \), the image of \( \alpha \in \Gamma(\pi(W), \mathbf{F}) \) works.  

1) \( \pi \circ (\pi|_W)^{-1}(q) = \pi(\pi^{-1}(q)) = q \) for all \( q \in \pi(W) \), hence indeed \( \alpha \in \Gamma(\pi(W), \mathbf{F}) \). 

2) As \( W \) was a neighborhood of \( p \), it follows that \( \alpha(x) = (\pi|_W)^{-1}(x) = p \). This is equivalent to saying that \( j_x(\alpha_x) = p \).

Injectivity: Suppose that \( j_x(\alpha_x) = j_x(\beta_x) = p \). By restricting the map to its intersection, we can find representatives of \( \alpha_x \) and \( \beta_x \) from same \( \Gamma(U, \mathbf{F}) \). Now pick \( W \) a neighborhood of \( p \) which induces a homeomorphism \( \pi|_W : W \rightarrow \pi(W) \). We may assume that \( W \subset U \) by replacing \( W \) by \( W \cap U \). We still get a homeomorphism because homeomorphisms are compatible with restrctions. We claim that \( \alpha \sim (\pi|_W)^{-1} \). 

Pick a point \( \pi(w) \in \pi(W) \). 

\[ \pi \circ \alpha(w) = w \]

by definition of a section then

\[ (\pi|_W)^{-1} \circ \pi \circ \alpha(w) = (\pi|_W)^{-1}(w) \]

Hence \( \alpha \) and \( (\pi|_W)^{-1} \) agrees on \( \pi(W) \). Their images are the same on stalk. Similarly, \( \beta \) also agrres on \( \pi(W) \), hence its image is the same with the image of \( (\pi|_W)^{-1} \). Namely, \( \alpha_x = \beta_x \).  

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Now our last task is to construct an ètale space from a presheaf \( \mathcal{F} \). We define

\[ \mathcal{F}^+ := \coprod_{x \in X} \mathcal{F}_x \]

where \( \coprod \) is disjoint union.

We also define

\[ \mathcal{W}(\alpha, U) := \{ \alpha_x ~|~ x \in U \} \text{ where } \alpha \in \mathcal{F}(U) \text{ and } U \in \Omega_x \]

We claim that \( \mathcal{W}(\alpha, U) \) form a base of a topology. Consider two such sets \( \mathcal{W}(\alpha, U) \) and \( \mathcal{W}(\beta, V) \). Let \( p \) be a point in the intersection, then there exists \( \alpha_x , \beta_x \) such that \( \alpha_x = p = \beta_x \). Because they are the same germ, we must have \( W \in \Omega_x \) such that \( \alpha|_W = \beta|_W \). \( W \) contains \( x \) because it must be a subset of the intersection of \( U \) and \( V \). Let \( \gamma = \alpha|_W = \beta|_W \), then \( p \in \mathcal{W}(\gamma, W) \subset \mathcal{W}(\alpha, U) \cap \mathcal{W}(\beta, V) \). 

The topology on \( \mathcal{F}^+ \) is defined to be the topology generated by \( \mathcal{W}(\alpha, U) \). Let \( \pi : \mathcal{F}^+ \rightarrow X \) be the map where \( \pi(p) = x \) if \( p \in \mathcal{F}_x \). This is well-defined because we had a disjoint union. Pick \( \alpha_x \in \mathcal{F}^+ \), then we have a homeomorphism between \( \mathcal{W}(\alpha, U)  \) and \( U = \pi(\mathcal{W})(\alpha, U) \). For any open subset \( V \subset U \), we have that \( \pi^{-1}(V) = \mathcal{W}(\alpha, V) \) which is also open. Any open subset of open set as a subspace topology is open in the original topological space. This proves that \( \pi \) is a local homeomorphism.

Hence the triple \( \mathcal{F}^+, \pi, X \) is an ètale space, and we denote the associated sheaf by \( \mathcal{F}^a \) instead of \( \mathcal{F}^{+b} \) for notational ease. 

For \( \alpha \in \mathcal{F}(U) \), we associate \( \alpha^+ \in \mathcal{F}^a(U) \) where \( \alpha^+(x) = \alpha_x \) for all \( x \in U \).  If  \( \mathcal{F} \) is a presheaf of algebraic structures (e.g. groups, rings, \( R \)-modules, etc. ), we define the structure of \( \mathcal{F}^a(U) \) in the obvious way such that the map \( \alpha \mapsto \alpha^+ \) is a homomorphism. 

Reference: [Iitaka] Algebraic Geometry (30-31)