# The Stacks Project

## Tag 00AA

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}

There are no comments yet for this tag.

There are also 2 comments on Section 6.31: Sheaves on Spaces.

## Add a comment on tag 00AA

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).