# The Stacks Project

## Tag 00A6

Lemma 6.31.7. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. Let $(\mathcal{C}, F)$ be a type of algebraic structure such that $\mathcal{C}$ has an initial object $e$. Consider the functors of restriction and extension by $e$ for (pre)sheaves of algebraic structure defined above.

1. The functor $j_{p!}$ is a left adjoint to the restriction functor $j_p$ (see Lemma 6.31.1).
2. The functor $j_!$ is a left adjoint to restriction, in a formula $$\mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(X, \mathcal{C})}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(U, \mathcal{C})}(\mathcal{F}, j^{-1}\mathcal{G}) = \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(U, \mathcal{C})}(\mathcal{F}, \mathcal{G}|_U)$$ bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
3. Let $\mathcal{F}$ be a sheaf on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows $$j_{!}\mathcal{F}_x = \left\{ \begin{matrix} e & \text{if} & x \not \in U \\ \mathcal{F}_x & \text{if} & x \in U \end{matrix} \right.$$
4. On the category of presheaves of algebraic structures on $U$ we have $j_pj_{p!} = \text{id}$.
5. On the category of sheaves of algebraic structures on $U$ we have $j^{-1}j_! = \text{id}$.

Proof. Omitted. $\square$

The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4654–4692 (see updates for more information).

\begin{lemma}
\label{lemma-j-shriek-structures}
Let $X$ be a topological space.
Let $j : U \to X$ be the inclusion of an open subset.
Let $(\mathcal{C}, F)$ be a type of algebraic structure
such that $\mathcal{C}$ has an initial object $e$.
Consider the functors of restriction and extension
by $e$ for (pre)sheaves of algebraic structure defined above.
\begin{enumerate}
\item The functor $j_{p!}$ is a left adjoint to the
restriction functor $j_p$ (see Lemma \ref{lemma-j-pullback}).
\item The functor $j_!$ is a left adjoint to restriction,
in a formula
$$\Mor_{\Sh(X, \mathcal{C})}(j_!\mathcal{F}, \mathcal{G}) = \Mor_{\Sh(U, \mathcal{C})}(\mathcal{F}, j^{-1}\mathcal{G}) = \Mor_{\Sh(U, \mathcal{C})}(\mathcal{F}, \mathcal{G}|_U)$$
bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
\item Let $\mathcal{F}$ be a sheaf on $U$.
The stalks of the sheaf $j_!\mathcal{F}$ are described
as follows
$$j_{!}\mathcal{F}_x = \left\{ \begin{matrix} e & \text{if} & x \not \in U \\ \mathcal{F}_x & \text{if} & x \in U \end{matrix} \right.$$
\item On the category of presheaves of algebraic structures on $U$
we have $j_pj_{p!} = \text{id}$.
\item On the category of sheaves of algebraic structures on $U$
we have $j^{-1}j_! = \text{id}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

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