85.16 The site associate to a simplicial semi-representable object
Let $\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\text{SR}(\mathcal{C})$. As usual, set $K_ n = K([n])$ and denote $K(\varphi ) : K_ n \to K_ m$ the morphism associated to $\varphi : [m] \to [n]$. By the construction in Section 85.15 we obtain a simplicial object $n \mapsto \mathcal{C}/K_ n$ in the category whose objects are sites and whose morphisms are cocontinuous functors. In other words, we get a gadget as in Case B of Section 85.3. The functors satisfy property P by Lemma 85.15.1. Hence we may apply Lemma 85.3.2 to obtain a site $(\mathcal{C}/K)_{total}$.
We can describe the site $(\mathcal{C}/K)_{total}$ explicitly as follows. Say $K_ n = \{ U_{n, i}\} _{i \in I_ n}$. For $\varphi : [m] \to [n]$ the morphism $K(\varphi ) : K_ n \to K_ m$ is given by a map $\alpha (\varphi ) : I_ n \to I_ m$ and morphisms $f_{\varphi , i} : U_{n, i} \to U_{m, \alpha (\varphi )(i)}$ for $i \in I_ n$. Then we have
an object of $(\mathcal{C}/K)_{total}$ corresponds to an object $(U/U_{n, i})$ of $\mathcal{C}/U_{n, i}$ for some $n$ and some $i \in I_ n$,
a morphism between $U/U_{n, i}$ and $V/U_{m, j}$ is a pair $(\varphi , f)$ where $\varphi : [m] \to [n]$, $j = \alpha (\varphi )(i)$, and $f : U \to V$ is a morphism of $\mathcal{C}$ such that
\[ \vcenter { \xymatrix{ U \ar[r]_ f \ar[d] & V \ar[d] \\ U_{n, i} \ar[r]^-{f_{\varphi , i}} & U_{m, j} } } \]
is commutative, and
coverings of the object $U/U_{n, i}$ are constructed by starting with a covering $\{ f_ j : U_ j \to U\} $ in $\mathcal{C}$ and letting $\{ (\text{id}, f_ j) : U_ j/U_{n, i} \to U/U_{n, i}\} $ be a covering in $(\mathcal{C}/K)_{total}$.
All of our general theory developed for simplicial sites applies to $(\mathcal{C}/K)_{total}$. Observe that the obvious forgetful functor
\[ j_{total} : (\mathcal{C}/K)_{total} \longrightarrow \mathcal{C} \]
is continuous and cocontinuous. It turns out that the associated morphism of topoi comes from an (obvious) augmentation.
Lemma 85.16.1. Let $\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\text{SR}(\mathcal{C})$. The localization functor $j_0 : \mathcal{C}/K_0 \to \mathcal{C}$ defines an augmentation $a_0 : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_0) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, as in case (B) of Remark 85.4.1. The corresponding morphisms of topoi
\[ a_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}),\quad a : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \]
of Lemma 85.4.2 are equal to the morphisms of topoi associated to the continuous and cocontinuous localization functors $j_ n : \mathcal{C}/K_ n \to \mathcal{C}$ and $j_{total} : (\mathcal{C}/K)_{total} \to \mathcal{C}$.
Proof.
This is immediate from working through the definitions. See in particular the footnote in the proof of Lemma 85.4.2 for the relationship between $a$ and $j_{total}$.
$\square$
Lemma 85.16.2. With assumption and notation as in Lemma 85.16.1 we have the following properties:
there is a functor $a^{Sh}_! : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ left adjoint to $a^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total})$,
there is a functor $a_! : \textit{Ab}((\mathcal{C}/K)_{total}) \to \textit{Ab}(\mathcal{C})$ left adjoint to $a^{-1} : \textit{Ab}(\mathcal{C}) \to \textit{Ab}((\mathcal{C}/K)_{total})$,
the functor $a^{-1}$ associates to $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ the sheaf on $(\mathcal{C}/K)_{total}$ which in degree $n$ is equal to $a_ n^{-1}\mathcal{F}$,
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 85.4.2. 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.
$\square$
Lemma 85.16.3. Let $\mathcal{C}$ be a site. Let $K$ be a simplicial object of $\text{SR}(\mathcal{C})$. Let $U/U_{n, i}$ be an object of $\mathcal{C}/K_ n$. Let $\mathcal{F} \in \textit{Ab}((\mathcal{C}/K)_{total})$. Then
\[ H^ p(U, \mathcal{F}) = H^ p(U, \mathcal{F}_{n, i}) \]
where
on the left hand side $U$ is viewed as an object of $\mathcal{C}_{total}$, and
on the right hand side $\mathcal{F}_{n, i}$ is the $i$th component of the sheaf $\mathcal{F}_ n$ on $\mathcal{C}/K_ n$ in the decomposition $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n) = \prod \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_{n, i})$ of Section 85.15.
Proof.
This follows immediately from Lemma 85.8.6 and the product decompositions of Section 85.15.
$\square$
All of the results of this section hold in this setting. To prove this one replaces the site $\mathcal{C}$ everywhere by $\mathcal{C}/X$.
The functor $a_!$ exists (but in general is not exact) because $a^{-1}\mathcal{O}_\mathcal {C} = \mathcal{O}$ and we can replace the use of Modules on Sites, Lemma 18.16.2 in the proof of Lemma 85.16.2 by Modules on Sites, Lemma 18.41.1. As discussed in Remark 85.15.6 there are exact functors $a_{n!} : \textit{Mod}(\mathcal{O}_ n) \to \textit{Mod}(\mathcal{O}_\mathcal {C})$ left adjoint to $a_ n^*$. Consequently, the morphisms $a$ and $a_ n$ are flat. Remark 85.15.6 implies the morphism of ringed topoi $f_\varphi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ m), \mathcal{O}_ m)$ for $\varphi : [m] \to [n]$ is flat and there exists an exact functor $f_{\varphi !} : \textit{Mod}(\mathcal{O}_ n) \to \textit{Mod}(\mathcal{O}_ m)$ left adjoint to $f_\varphi ^*$. This in turn implies that for the flat morphism of ringed topoi $g_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}), \mathcal{O})$ the functor $g_{n!} : \textit{Mod}(\mathcal{O}_ n) \to \textit{Mod}(\mathcal{O})$ left adjoint to $g_ n^*$ is exact, see Lemma 85.6.3.
Of course, all the results mentioned in Remark 85.16.5 hold in this setting as well.
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