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Étalé space (Definition)

The Étalé space (Espace Étalé) is a topological space associated to a presheaf $ \ensuremath{\mathcal{F}}$ on a space $ X$. The Étalé space is defined to be the disjoint union of stalks of the sheaf $ \ensuremath{\mathcal{F}}$.

$\displaystyle \ensuremath{\mathcal{E}}_{\ensuremath{\mathcal{F}}} \equiv \coprod_{x\in X} \ensuremath{\mathcal{F}}_x$

Over each open set $ U\subset X$, there is a set of sections $ \Gamma(U,\ensuremath{\mathcal{F}})$. A basis for the topology on the Étalé space is formed by taking the open sets to be of the form $ \ensuremath{\mathcal{U}}_s = \{s_x, x\in U\}$, for $ s\in \Gamma(U,\ensuremath{\mathcal{F}})$ and $ s_x$ the germ of $ s$ at $ x$. There is a natural map $ \pi\!:\!\ensuremath{\mathcal{E}}_{\ensuremath{\mathcal{F}}} \rightarrow X$ which takes germs $ s_x$ in the stalk $ \ensuremath{\mathcal{F}}_x$ over $ x$ to $ x$.

Let $ s\in \Gamma(U,\ensuremath{\mathcal{F}})$ and $ s^\prime \in \Gamma(U^\prime,\ensuremath{\mathcal{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\vert _V = s^\prime\vert _V$). This verifies that $ \{\ensuremath{\mathcal{U}}_s\}$ form a basis for $ \ensuremath{\mathcal{E}}_{\ensuremath{\mathcal{F}}}$.

Then there is another presheaf, $ \widetilde{\ensuremath{\mathcal{F}}}$, whose sections are the continuous functions from $ X$ to $ \ensuremath{\mathcal{E}}_{\ensuremath{\mathcal{F}}}$. This presheaf forms a sheaf, and is equivalent to the sheafification of the presheaf $ \ensuremath{\mathcal{F}}$.



"Étalé space" is owned by guffin.
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See Also: stalk, sheaf

Other names:  Espace Etale, Etale space, Espace Étalé
Also defines:  Étalé Space, Etale Space
Keywords:  Sheaf, Stalk, Etale Space, Sheafification
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Cross-references: sheafification, equivalent, continuous functions, point, map, germ, basis, sections, open set, sheaf, stalks, disjoint union, presheaf, topological space, espace
There are 3 references to this entry.

This is version 1 of Étalé space, born on 2006-02-08.
Object id is 7608, canonical name is EtaleSpace.
Accessed 3378 times total.

Classification:
AMS MSC: 14F05 (Algebraic geometry :: homology theory :: Vector bundles, sheaves, related constructions)
 54B40 (General topology :: Basic constructions :: Presheaves and sheaves)
 18F20 (Category theory; homological algebra :: Categories and geometry :: Presheaves and sheaves)
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