## 85.21 Hypercovering by a simplicial object of the site

Let $\mathcal{C}$ be a site with fibre products and let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. In this section we elucidate the results of Section 85.19 in the case that our hypercovering is given by a simplicial object of the site. Let $U$ be a simplicial object of $\mathcal{C}$. As usual we denote $U_ n = U([n])$ and $f_\varphi : U_ n \to U_ m$ the morphism $f_\varphi = U(\varphi )$ corresponding to $\varphi : [m] \to [n]$. Assume we have an augmentation

$a : U \to X$

From this we obtain a simplicial site $(\mathcal{C}/U)_{total}$ and an augmentation morphism

$a : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/U)_{total}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X)$

Namely, from $U$ we obtain a simiplical object $K$ of $\text{SR}(\mathcal{C}, X)$ with degree $n$ part $K_ n = \{ U_ n \to X\}$ and we can apply the constructions in Remark 85.16.4. More precisely, an object of the site $(\mathcal{C}/U)_{total}$ is given by a $V/U_ n$ and a morphism $(\varphi , f) : V/U_ n \to W/U_ m$ is given by a morphism $\varphi : [m] \to [n]$ in $\Delta$ and a morphism $f : V \to W$ such that the diagram

$\xymatrix{ V \ar[r]_ f \ar[d] & W \ar[d] \\ U_ n \ar[r]^{f_\varphi } & U_ m }$

is commutative. The morphism of topoi $a$ is given by the cocontinuous functor $V/U_ n \mapsto V/X$. That's all folks!

In this section we will say the augmentation $a : U \to X$ is a hypercovering of $X$ in $\mathcal{C}$ if the following hold

1. $\{ U_0 \to X\}$ is a covering of $\mathcal{C}$,

2. $\{ U_1 \to U_0 \times _ X U_0\}$ is a covering of $\mathcal{C}$,

3. $\{ U_{n + 1} \to (\text{cosk}_ n\text{sk}_ n U)_{n + 1}\}$ is a covering of $\mathcal{C}$ for $n \geq 1$.

This is equivalent to the condition that $K$ (as above) is a hypercovering of $X$, see Hypercoverings, Example 25.3.5.

Lemma 85.21.1. Let $\mathcal{C}$ be a site with fibre product and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above. Then

1. $a^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X) \to \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/U)_{total})$ is fully faithful with essential image the cartesian sheaves of sets,

2. $a^{-1} : \textit{Ab}(\mathcal{C}/X) \to \textit{Ab}((\mathcal{C}/U)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups.

In both cases $a_*$ provides the quasi-inverse functor.

Proof. This is a special case of Lemma 85.19.1. $\square$

Lemma 85.21.2. Let $\mathcal{C}$ be a site with fibre product and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above. For $E \in D(\mathcal{C}/X)$ the map

$E \longrightarrow Ra_*a^{-1}E$

is an isomorphism.

Proof. This is a special case of Lemma 85.19.2. $\square$

Lemma 85.21.3. Let $\mathcal{C}$ be a site with fibre products and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above. Then we have a canonical isomorphism

$R\Gamma (X, E) = R\Gamma ((\mathcal{C}/U)_{total}, a^{-1}E)$

for $E \in D(\mathcal{C}/X)$.

Proof. This is a special case of Lemma 85.19.3. $\square$

Lemma 85.21.4. Let $\mathcal{C}$ be a site with fibre product and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $a : U \to X$ be a hypercovering of $X$ in $\mathcal{C}$ as defined above. Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/U)_{total})$ denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor $a^{-1}$ defines an equivalence

$D^+(\mathcal{C}/X) \longrightarrow D_\mathcal {A}^+((\mathcal{C}/U)_{total})$

with quasi-inverse $Ra_*$.

Proof. This is a special case of Lemma 85.19.4 $\square$

Lemma 85.21.5. Let $U$ be a simplicial object of a site $\mathcal{C}$ with fibre products.

1. $\mathcal{C}/U$ has the structure of a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites,

2. the construction of Lemma 85.3.1 applied to the structure in (1) reproduces the site $(\mathcal{C}/U)_{total}$ above,

3. if $a : U \to X$ is an augmentation, then $a_0 : \mathcal{C}/U_0 \to \mathcal{C}/X$ is an augmentation as in Remark 85.4.1 part (A) and gives the same morphism of topoi $a : \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/U)_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X)$ as the one above.

Proof. Given a morphism of objects $V \to W$ of $\mathcal{C}$ the localization morphism $j : \mathcal{C}/V \to \mathcal{C}/W$ is a left adjoint to the base change functor $\mathcal{C}/W \to \mathcal{C}/V$. The base change functor is continuous and induces the same morphism of topoi as $j$. See Sites, Lemma 7.27.3. This proves (1).

Part (2) holds because a morphism $V/U_ n \to W/U_ m$ of the category constructed in Lemma 85.3.1 is a morphism $V \to W \times _{U_ m, f_\varphi } U_ n$ over $U_ n$ which is the same thing as a morphism $f : V \to W$ over the morphism $f_\varphi : U_ n \to U_ m$, i.e., the same thing as a morphism in the category $(\mathcal{C}/U)_{total}$ defined above. Equality of sets of coverings is immediate from the definition.

We omit the proof of (3). $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).