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Tag 03EH

Chapter 7: Sites and Sheaves > Section 7.24: Localization

Lemma 7.24.8. Let $\mathcal{C}$ be a site. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Then there exists a commutative diagram $$ \xymatrix{ \mathcal{C}/V \ar[rd]_{j_V} \ar[rr]_j & & \mathcal{C}/U \ar[ld]^{j_U} \\ & \mathcal{C} & } $$ of cocontinuous functors. Here $j : \mathcal{C}/V \to \mathcal{C}/U$, $(a : W \to V) \mapsto (f \circ a : W \to U)$ is identified with the functor $j_{V/U} : (\mathcal{C}/U)/(V/U) \to \mathcal{C}/U$ via the identification $(\mathcal{C}/U)/(V/U) = \mathcal{C}/V$. Moreover we have $j_{V!} = j_{U!} \circ j_!$, $j_V^{-1} = j^{-1} \circ j_U^{-1}$, and $j_{V*} = j_{U*} \circ j_*$.

Proof. The commutativity of the diagram is immediate. The agreement of $j$ with $j_{V/U}$ follows from the definitions. By Lemma 7.20.2 we see that the following diagram of morphisms of topoi \begin{equation} \tag{7.24.8.1} \vcenter{ \xymatrix{ \mathop{\textit{Sh}}\nolimits(\mathcal{C}/V) \ar[rd]_{j_V} \ar[rr]_j & & \mathop{\textit{Sh}}\nolimits(\mathcal{C}/U) \ar[ld]^{j_U} \\ & \mathop{\textit{Sh}}\nolimits(\mathcal{C}) & } } \end{equation} is commutative. This proves that $j_V^{-1} = j^{-1} \circ j_U^{-1}$ and $j_{V*} = j_{U*} \circ j_*$. The equality $j_{V!} = j_{U!} \circ j_!$ follows formally from adjointness properties. $\square$

    The code snippet corresponding to this tag is a part of the file sites.tex and is located in lines 4870–4889 (see updates for more information).

    \begin{lemma}
    \label{lemma-relocalize}
    Let $\mathcal{C}$ be a site.
    Let $f : V \to U$ be a morphism of $\mathcal{C}$.
    Then there exists a commutative diagram
    $$
    \xymatrix{
    \mathcal{C}/V \ar[rd]_{j_V} \ar[rr]_j & &
    \mathcal{C}/U \ar[ld]^{j_U} \\
    & \mathcal{C} &
    }
    $$
    of cocontinuous functors. Here $j : \mathcal{C}/V \to \mathcal{C}/U$,
    $(a : W \to V) \mapsto (f \circ a : W \to U)$
    is identified with the functor
    $j_{V/U} : (\mathcal{C}/U)/(V/U) \to \mathcal{C}/U$
    via the identification $(\mathcal{C}/U)/(V/U) = \mathcal{C}/V$.
    Moreover we have $j_{V!} = j_{U!} \circ j_!$,
    $j_V^{-1} = j^{-1} \circ j_U^{-1}$, and $j_{V*} = j_{U*} \circ j_*$.
    \end{lemma}
    
    \begin{proof}
    The commutativity of the diagram is immediate.
    The agreement of $j$ with $j_{V/U}$ follows from the definitions. By
    Lemma \ref{lemma-composition-cocontinuous}
    we see that the following diagram of morphisms of topoi
    \begin{equation}
    \label{equation-relocalize}
    \vcenter{
    \xymatrix{
    \Sh(\mathcal{C}/V) \ar[rd]_{j_V} \ar[rr]_j & &
    \Sh(\mathcal{C}/U) \ar[ld]^{j_U} \\
    & \Sh(\mathcal{C}) &
    }
    }
    \end{equation}
    is commutative. This proves that
    $j_V^{-1} = j^{-1} \circ j_U^{-1}$ and $j_{V*} = j_{U*} \circ j_*$.
    The equality $j_{V!} = j_{U!} \circ j_!$
    follows formally from adjointness properties.
    \end{proof}

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