# The Stacks Project

## Tag 00A2

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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