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.

The functor $j_{p!}$ is a left adjoint to the restriction functor $j_ p$ (see Lemma 6.31.1).

The functor $j_!$ is a left adjoint to restriction, in a formula

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X, \mathcal{C})}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (U, \mathcal{C})}(\mathcal{F}, j^{-1}\mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (U, \mathcal{C})}(\mathcal{F}, \mathcal{G}|_ U) \]bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.

Let $\mathcal{F}$ be a sheaf on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows

\[ j_{!}\mathcal{F}_ x = \left\{ \begin{matrix} e & \text{if} & x \not\in U \\ \mathcal{F}_ x & \text{if} & x \in U \end{matrix} \right. \]On the category of presheaves of algebraic structures on $U$ we have $j_ pj_{p!} = \text{id}$.

On the category of sheaves of algebraic structures on $U$ we have $j^{-1}j_! = \text{id}$.

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