## 84.16 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 84.15 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 84.3. The functors satisfy property P by Lemma 84.15.1. Hence we may apply Lemma 84.3.2 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

1. 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 $i \in I_ n$,

2. 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

3. 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 84.16.1. 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 84.4.1. 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 84.4.2 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 84.4.2 for the relationship between $a$ and $j_{total}$. $\square$

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. $\square$

Lemma 84.16.3. 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

1. on the left hand side $U$ is viewed as an object of $\mathcal{C}_{total}$, and

2. 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 84.15.

Proof. This follows immediately from Lemma 84.8.6 and the product decompositions of Section 84.15. $\square$

Remark 84.16.4 (Variant for over an object). Let $\mathcal{C}$ be a site. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Recall that we have a category $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$ of semi-representable objects over $X$, see Remark 84.15.5. We may apply the above discussion to the site $\mathcal{C}/X$. Briefly, the constructions above give

1. a site $(\mathcal{C}/K)_{total}$ for a simplicial $K$ object of $\text{SR}(\mathcal{C}, X)$,

2. a localization functor $j_{total} : (\mathcal{C}/K)_{total} \to \mathcal{C}/X$,

3. localization functors $j_ n : \mathcal{C}/K_ n \to \mathcal{C}/X$,

4. a morphism of topoi $a : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X)$,

5. morphisms of topoi $a_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X)$,

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

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

All of the results of this section hold in this setting. To prove this one replaces the site $\mathcal{C}$ everywhere by $\mathcal{C}/X$.

Remark 84.16.5 (Ringed variant). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Given a simplicial semi-representable object $K$ of $\mathcal{C}$ we set $\mathcal{O} = a^{-1}\mathcal{O}_\mathcal {C}$, where $a$ is as in Lemmas 84.16.1 and 84.16.2. The constructions above, keeping track of the sheaves of rings as in Remark 84.15.6, give

1. a ringed site $((\mathcal{C}/K)_{total}, \mathcal{O})$ for a simplicial $K$ object of $\text{SR}(\mathcal{C})$,

2. a morphism of ringed topoi $a : (\mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$,

3. morphisms of ringed topoi $a_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$,

4. a functor $a_! : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_\mathcal {C})$ left adjoint to $a^*$.

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 84.16.2 by Modules on Sites, Lemma 18.41.1. As discussed in Remark 84.15.6 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 84.15.6 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 84.6.3.

Remark 84.16.6 (Ringed variant over an object). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and denote $\mathcal{O}_ X = \mathcal{O}_\mathcal {C}|_{\mathcal{C}/X}$. Then we can combine the constructions given in Remarks 84.16.4 and 84.16.5 to get

1. a ringed site $((\mathcal{C}/K)_{total}, \mathcal{O})$ for a simplicial $K$ object of $\text{SR}(\mathcal{C}, X)$,

2. a morphism of ringed topoi $a : (\mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X), \mathcal{O}_ X)$,

3. morphisms of ringed topoi $a_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X), \mathcal{O}_ X)$,

4. a functor $a_! : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ X)$ left adjoint to $a^*$.

Of course, all the results mentioned in Remark 84.16.5 hold in this setting as well.

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