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Tag 00A4

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

Definition 6.31.5. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.

  1. Let $\mathcal{F}$ be an abelian presheaf on $U$. We define the extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $0$ to be the abelian presheaf on $X$ defined by the rule $$ j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} 0 & \text{if} & V \not \subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right. $$ with obvious restriction mappings.
  2. Let $\mathcal{F}$ be an abelian sheaf on $U$. We define the extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $0$ to be the sheafification of the abelian presheaf $j_{p!}\mathcal{F}$.
  3. Let $\mathcal{C}$ be a category having an initial object $e$. Let $\mathcal{F}$ be a presheaf on $U$ with values in $\mathcal{C}$. We define the extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $e$ to be the presheaf on $X$ with values in $\mathcal{C}$ defined by the rule $$ j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} e & \text{if} & V \not \subset U \\ \mathcal{F}(V) & \text{if} & V \subset U \end{matrix} \right. $$ with obvious restriction mappings.
  4. Let $(\mathcal{C}, F)$ be a type of algebraic structure such that $\mathcal{C}$ has an initial object $e$. Let $\mathcal{F}$ be a sheaf of algebraic structures on $U$ (of the give type). We define the extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $e$ to be the sheafification of the presheaf $j_{p!}\mathcal{F}$ defined above.
  5. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}|_U$-modules. In this case we define the extension by $0$ to be the presheaf of $\mathcal{O}$-modules which is equal to $j_{p!}\mathcal{F}$ as an abelian presheaf endowed with the multiplication map $\mathcal{O} \times j_{p!}\mathcal{F} \to j_{p!}\mathcal{F}$.
  6. Let $\mathcal{O}$ be a sheaf of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}|_U$-modules. In this case we define the extension by $0$ to be the $\mathcal{O}$-module which is equal to $j_!\mathcal{F}$ as an abelian sheaf endowed with the multiplication map $\mathcal{O} \times j_!\mathcal{F} \to j_!\mathcal{F}$.

    The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4539–4596 (see updates for more information).

    \begin{definition}
    \label{definition-j-shriek-structures}
    Let $X$ be a topological space.
    Let $j : U \to X$ be the inclusion of an open subset.
    \begin{enumerate}
    \item Let $\mathcal{F}$ be an abelian presheaf on $U$.
    We define the {\it extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $0$}
    to be the abelian presheaf on $X$ defined by the rule
    $$
    j_{p!}\mathcal{F}(V) =
    \left\{
    \begin{matrix}
    0 & \text{if} & V \not \subset U \\
    \mathcal{F}(V) & \text{if} & V \subset U
    \end{matrix}
    \right.
    $$
    with obvious restriction mappings.
    \item Let $\mathcal{F}$ be an abelian sheaf on $U$. We define
    the {\it extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $0$}
    to be the sheafification of the abelian presheaf $j_{p!}\mathcal{F}$.
    \item Let $\mathcal{C}$ be a category having an initial object $e$.
    Let $\mathcal{F}$ be a presheaf on $U$ with values in $\mathcal{C}$.
    We define the {\it extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $e$}
    to be the presheaf on $X$ with values in $\mathcal{C}$ defined by the
    rule
    $$
    j_{p!}\mathcal{F}(V) =
    \left\{
    \begin{matrix}
    e & \text{if} & V \not \subset U \\
    \mathcal{F}(V) & \text{if} & V \subset U
    \end{matrix}
    \right.
    $$
    with obvious restriction mappings.
    \item Let $(\mathcal{C}, F)$ be a type of algebraic structure
    such that $\mathcal{C}$ has an initial object $e$.
    Let $\mathcal{F}$ be a sheaf of algebraic structures on $U$
    (of the give type). We define the
    {\it extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $e$}
    to be the sheafification of the presheaf $j_{p!}\mathcal{F}$
    defined above.
    \item Let $\mathcal{O}$ be a presheaf of rings on $X$.
    Let $\mathcal{F}$ be a presheaf of $\mathcal{O}|_U$-modules.
    In this case we define the {\it extension by $0$}
    to be the presheaf of $\mathcal{O}$-modules which is equal to
    $j_{p!}\mathcal{F}$ as an abelian presheaf endowed with
    the multiplication map
    $\mathcal{O} \times j_{p!}\mathcal{F} \to j_{p!}\mathcal{F}$.
    \item Let $\mathcal{O}$ be a sheaf of rings on $X$.
    Let $\mathcal{F}$ be a sheaf of $\mathcal{O}|_U$-modules.
    In this case we define the {\it extension by $0$}
    to be the $\mathcal{O}$-module which is equal to
    $j_!\mathcal{F}$ as an abelian sheaf endowed with
    the multiplication map $\mathcal{O} \times j_!\mathcal{F} \to j_!\mathcal{F}$.
    \end{enumerate}
    \end{definition}

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