(Espace étalé)
Let \(\mcf\) be a presheaf on \(X\). Set \[ \Et(\mcf) \da \disjoint_{p \in X} \mcf_p , \qquad \pi : \Et(\mcf) \to X , \] where \(\pi\) sends every germ in \(\mcf_p\) to \(p\).
Each \(s \in \mcf(U)\) gives a set-theoretic section \(\bar s : U \to \Et(\mcf)\), \(p \mapsto s_p\), with \(\pi \circ \bar s = \id_U\). Give \(\Et(\mcf)\) the finest topology making every such \(\bar s\) continuous.
For that topology, \[ \mcf^+(U) = \ts{ \text{continuous } t : U \to \Et(\mcf) \st \pi \circ t = \id_U } . \]
The sets \(\bar s(U)\), for \(U\) open and \(s \in \mcf(U)\), form a basis of the topology, and each \(\bar s\) is a homeomorphism of \(U\) onto the open set \(\bar s(U)\), with inverse \(\pi|_{\bar s(U)}\). Hence \(\pi\) is a local homeomorphism.
(Sheaves as spaces over $X$)
In T-3VX80, \(\mcf^+(U)\) is the set of compatible families of germs; the proposition identifies these with the continuous sections of \(\pi\). The functor \(\mcf\mapsto(\Et(\mcf)\to X)\) restricts to an equivalence between sheaves on \(X\) and spaces over \(X\) whose projection is a local homeomorphism; the inverse sends such a space to its sheaf of continuous sections.
Since \(\Et(\mcf)\) is built from the stalks, \((\mcf^+)_p = \mcf_p\) for every \(p\). The map \(\mcf \to \mcf^+\) is an isomorphism if and only if \(\mcf\) is a sheaf.
For an abelian group \(A\), the constant presheaf \(U\mapsto A\) and the constant sheaf \(\ul{A}\) have the same espace étalé, \(X \times A\) with \(A\) discrete; its continuous sections over \(U\) are the locally constant functions \(U\to A\), which are the sections of \(\ul{A}\).
Dependencies
D-0QSI0Stalks and germsuses
Backlinks
FE-SHFISOSTALKSSheaves with isomorphic stalks, and pushforwards of locally constant sheavesuses