The Stacks project

Lemma 84.16.2. With assumption and notation as in Lemma 84.16.1 we have the following properties:

  1. there is a functor $a^{Sh}_! : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ left adjoint to $a^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total})$,

  2. there is a functor $a_! : \textit{Ab}((\mathcal{C}/K)_{total}) \to \textit{Ab}(\mathcal{C})$ left adjoint to $a^{-1} : \textit{Ab}(\mathcal{C}) \to \textit{Ab}((\mathcal{C}/K)_{total})$,

  3. the functor $a^{-1}$ associates to $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ the sheaf on $(\mathcal{C}/K)_{total}$ wich in degree $n$ is equal to $a_ n^{-1}\mathcal{F}$,

  4. the functor $a_*$ associates to $\mathcal{G}$ in $\textit{Ab}((\mathcal{C}/K)_{total})$ the equalizer of the two maps $j_{0, *}\mathcal{G}_0 \to j_{1, *}\mathcal{G}_1$,

Proof. Parts (3) and (4) hold for any augmentation of a simplicial site, see Lemma 84.4.2. Parts (1) and (2) follow as $j_{total}$ is continuous and cocontinuous. The functor $a^{Sh}_!$ is constructed in Sites, Lemma 7.21.5 and the functor $a_!$ is constructed in Modules on Sites, Lemma 18.16.2. $\square$

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