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.
$\square$
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