Sheaf associated to a Presheaf « Taggable Mathematics

Sheaf associated to a Presheaf

November 3, 2009 – 10:26 pm
Posted in Proposition, definition, math.AG, unfinished
Tagged construction, disjoint union, presheaf, sheaf, universal

Definition. Let $\mathscr F$ be a presheaf. We define the sheaf associated to $\mathscr F$ , denoted $\mathscr F^+$ , by associating $U\subseteq X$ open with the set of maps $s: U \rightarrow \bigsqcup_x \mathscr F_x$ such that

$s(x) \in \mathscr F_x$ for all $x \in U$ ;
For all $x \in U$ , there exists an open neighborhood $W\subseteq U$ of $x$ and $\alpha \in \mathscr F(U)$ such that $s(y) = \alpha_y$ for all $y \in W$ .

Proposition. The correspondence $\mathscr F^+$ is a sheaf.

Proof. To do!

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Sheaf associated to a Presheaf

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