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Tag 00A4

Definition 6.31.5. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.

1. Let $\mathcal{F}$ be an abelian presheaf on $U$. We define the extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $0$ to be the abelian presheaf on $X$ defined by the rule $$j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} 0 & \text{if} & V \not \subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right.$$ with obvious restriction mappings.
2. Let $\mathcal{F}$ be an abelian sheaf on $U$. We define the extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $0$ to be the sheafification of the abelian presheaf $j_{p!}\mathcal{F}$.
3. Let $\mathcal{C}$ be a category having an initial object $e$. Let $\mathcal{F}$ be a presheaf on $U$ with values in $\mathcal{C}$. We define the extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $e$ to be the presheaf on $X$ with values in $\mathcal{C}$ defined by the rule $$j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} e & \text{if} & V \not \subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right.$$ with obvious restriction mappings.
4. Let $(\mathcal{C}, F)$ be a type of algebraic structure such that $\mathcal{C}$ has an initial object $e$. Let $\mathcal{F}$ be a sheaf of algebraic structures on $U$ (of the give type). We define the extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $e$ to be the sheafification of the presheaf $j_{p!}\mathcal{F}$ defined above.
5. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}|_U$-modules. In this case we define the extension by $0$ to be the presheaf of $\mathcal{O}$-modules which is equal to $j_{p!}\mathcal{F}$ as an abelian presheaf endowed with the multiplication map $\mathcal{O} \times j_{p!}\mathcal{F} \to j_{p!}\mathcal{F}$.
6. Let $\mathcal{O}$ be a sheaf of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}|_U$-modules. In this case we define the extension by $0$ to be the $\mathcal{O}$-module which is equal to $j_!\mathcal{F}$ as an abelian sheaf endowed with the multiplication map $\mathcal{O} \times j_!\mathcal{F} \to j_!\mathcal{F}$.

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\begin{definition}
\label{definition-j-shriek-structures}
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 an abelian presheaf on $U$.
We define the {\it extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $0$}
to be the abelian presheaf on $X$ defined by the rule
$$j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} 0 & \text{if} & V \not \subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right.$$
with obvious restriction mappings.
\item Let $\mathcal{F}$ be an abelian sheaf on $U$. We define
the {\it extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $0$}
to be the sheafification of the abelian presheaf $j_{p!}\mathcal{F}$.
\item Let $\mathcal{C}$ be a category having an initial object $e$.
Let $\mathcal{F}$ be a presheaf on $U$ with values in $\mathcal{C}$.
We define the {\it extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $e$}
to be the presheaf on $X$ with values in $\mathcal{C}$ defined by the
rule
$$j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} e & \text{if} & V \not \subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right.$$
with obvious restriction mappings.
\item Let $(\mathcal{C}, F)$ be a type of algebraic structure
such that $\mathcal{C}$ has an initial object $e$.
Let $\mathcal{F}$ be a sheaf of algebraic structures on $U$
(of the give type). We define the
{\it extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $e$}
to be the sheafification of the presheaf $j_{p!}\mathcal{F}$
defined above.
\item Let $\mathcal{O}$ be a presheaf of rings on $X$.
Let $\mathcal{F}$ be a presheaf of $\mathcal{O}|_U$-modules.
In this case we define the {\it extension by $0$}
to be the presheaf of $\mathcal{O}$-modules which is equal to
$j_{p!}\mathcal{F}$ as an abelian presheaf endowed with
the multiplication map
$\mathcal{O} \times j_{p!}\mathcal{F} \to j_{p!}\mathcal{F}$.
\item Let $\mathcal{O}$ be a sheaf of rings on $X$.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}|_U$-modules.
In this case we define the {\it extension by $0$}
to be the $\mathcal{O}$-module which is equal to
$j_!\mathcal{F}$ as an abelian sheaf endowed with
the multiplication map $\mathcal{O} \times j_!\mathcal{F} \to j_!\mathcal{F}$.
\end{enumerate}
\end{definition}

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