The Stacks project

Lemma 84.15.3. Let $\mathcal{C}$ be a site.

  1. For $K$ in $\text{SR}(\mathcal{C})$ the functor $j_!$ gives an equivalence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(K)^\# $ where $F$ is as in Hypercoverings, Definition 25.2.2.

  2. The functor $j^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K)$ corresponds via the identification of (1) with $\mathcal{F} \mapsto (\mathcal{F} \times F(K)^\# \to F(K)^\# )$.

  3. For $f : K \to L$ in $\text{SR}(\mathcal{C})$ the functor $f^{-1}$ corresponds via the identifications of (1) to the functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(L)^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(K)^\# $, $(\mathcal{G} \to F(L)^\# ) \mapsto (\mathcal{G} \times _{F(L)^\# } F(K)^\# \to F(K)^\# )$.

Proof. Observe that if $K = \{ U_ i\} _{i \in I}$ then the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K)$ decomposes as the product of the categories $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$. Observe that $F(K)^\# = \coprod _{i \in I} h_{U_ i}^\# $ (coproduct in sheaves). Hence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(K)^\# $ is the product of the categories $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_{U_ i}^\# $. Thus (1) and (2) follow from the corresponding statements for each $i$, see Sites, Lemmas 7.25.4 and 7.25.7. Similarly, if $L = \{ V_ j\} _{j \in J}$ and $f$ is given by $\alpha : I \to J$ and $f_ i : U_ i \to V_{\alpha (i)}$, then we can apply Sites, Lemma 7.25.9 to each of the re-localization morphisms $\mathcal{C}/U_ i \to \mathcal{C}/V_{\alpha (i)}$ to get (3). $\square$

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