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stalk of a presheaf

DEFINITION 1.32   Let $ \mathcal G$ be a presheaf on a topological space $ X$ . Let $ P\in X$ be a point. We define the stalk of $ \mathcal G$ on $ P$ as

$\displaystyle \mathcal G_P=\varinjlim_{U\ni P} \mathcal G(U)
$

It should be noted at this stage that

LEMMA 1.33   Let $ \mathcal G$ be a presheaf on a topological space $ X$ . The natural map

$\displaystyle \mathcal G \to \operatorname{sheaf}(\mathcal G)
$

induces an isomorphism of stalk at each point $ x \in X$ .



2007-12-11