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Tag 00A2

Chapter 6: Sheaves on Spaces > Section 6.31: Open immersions and (pre)sheaves

Definition 6.31.3. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.

  1. Let $\mathcal{F}$ be a presheaf of sets on $U$. We define the extension of $\mathcal{F}$ by the empty set $j_{p!}\mathcal{F}$ to be the presheaf of sets on $X$ defined by the rule $$ j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} \emptyset & \text{if} & V \not \subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right. $$ with obvious restriction mappings.
  2. Let $\mathcal{F}$ be a sheaf of sets on $U$. We define the extension of $\mathcal{F}$ by the empty set $j_!\mathcal{F}$ to be the sheafification of the presheaf $j_{p!}\mathcal{F}$.

    The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4452–4474 (see updates for more information).

    \begin{definition}
    \label{definition-j-shriek}
    Let $X$ be a topological space.
    Let $j : U \to X$ be the inclusion of an open subset.
    \begin{enumerate}
    \item Let $\mathcal{F}$ be a presheaf of sets on $U$. We define
    the {\it extension of $\mathcal{F}$ by the empty set $j_{p!}\mathcal{F}$}
    to be the presheaf of sets on $X$ defined by the rule
    $$
    j_{p!}\mathcal{F}(V) =
    \left\{
    \begin{matrix}
    \emptyset & \text{if} & V \not \subset U \\
    \mathcal{F}(V) & \text{if} & V \subset U
    \end{matrix}
    \right.
    $$
    with obvious restriction mappings.
    \item Let $\mathcal{F}$ be a sheaf of sets on $U$. We define
    the {\it extension of $\mathcal{F}$ by the empty set $j_!\mathcal{F}$}
    to be the sheafification of the presheaf $j_{p!}\mathcal{F}$.
    \end{enumerate}
    \end{definition}

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