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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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