\'Etal\'e spaceEtaleSpace2006-02-08 11:37:352008-05-26 03:03:23Definition\'Etal\'e SpaceEtale SpaceSheafStalkEtale SpaceSheafification% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe \Etale~space (Espace \Etale) is a topological space associated to a presheaf $\sheaf F$ on a space $X$. The \'Etal\'e space is defined to be the disjoint union of stalks of the sheaf $\sheaf F$.
\[\sheaf E_{\sheaf F} \equiv \coprod_{x\in X} \sheaf F_x\]
Over each open set $U\subset X$, there is a set of sections $\Gamma(U,\sheaf F)$. A basis for the topology on the \Etale~space is formed by taking the open sets to be of the form $\sheaf U_s = \{s_x, x\in U\}$, for $s\in \Gamma(U,\sheaf F)$ and $s_x$ the germ of $s$ at $x$. There is a natural map $\pi\!:\!\sheaf E_{\sheaf F} \rightarrow X$ which takes germs $s_x$ in the stalk $\sheaf F_x$ over $x$ to $x$.
Let $s\in \Gamma(U,\sheaf F)$ and $s^\prime \in \Gamma(U^\prime,\sheaf F)$ with $U\cap U^\prime \ne \emptyset$. At each point $x\in U \cap U^\prime$ where $s_x = s^\prime_x$, by the definition of germs there exists an open set $V\subset U\cap U^\prime$ containing $x$ such that $s$ and $s^\prime$ restrict to the same section on $V$ ($s|_V = s^\prime|_V$). This verifies that $\{\sheaf U_s\}$ form a basis for $\sheaf E_{\sheaf F}$.
Then there is another presheaf, $\widetilde{\sheaf F}$, whose sections are the continuous functions from $X$ to $\sheaf E_{\sheaf F}$ assigning an element $s(x)\in \sheaf F_x$ to each point $x \in X$. This presheaf forms a sheaf equivalent to the sheafification of the presheaf $\sheaf F$.