## Tag `00AA`

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

Lemma 6.31.11. 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 $e$. The functor $$ j_! : \mathop{\textit{Sh}}\nolimits(U, \mathcal{C}) \longrightarrow \mathop{\textit{Sh}}\nolimits(X, \mathcal{C}) $$ is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_x = e$ for all $x \in X \setminus U$.

Proof.Omitted. $\square$

The code snippet corresponding to this tag is a part of the file `sheaves.tex` and is located in lines 4793–4806 (see updates for more information).

```
\begin{lemma}
\label{lemma-equivalence-categories-open-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 $e$.
The functor
$$
j_! : \Sh(U, \mathcal{C}) \longrightarrow \Sh(X, \mathcal{C})
$$
is fully faithful. Its essential image consists exactly
of those sheaves $\mathcal{G}$ such that
$\mathcal{G}_x = e$ for all $x \in X \setminus U$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
```

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