## 84.21 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 84.19 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 84.16.4. 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

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

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

$\{ 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 84.21.1. 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

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

$a^{-1} : \textit{Ab}(\mathcal{C}/X) \to \textit{Ab}((\mathcal{C}/U)_{total})$ 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 84.19.1.
$\square$

Lemma 84.21.2. 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 84.19.2.
$\square$

Lemma 84.21.3. 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 84.19.3.
$\square$

Lemma 84.21.4. 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 84.19.4
$\square$

Lemma 84.21.5. Let $U$ be a simplicial object of a site $\mathcal{C}$ with fibre products.

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

the construction of Lemma 84.3.1 applied to the structure in (1) reproduces the site $(\mathcal{C}/U)_{total}$ above,

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 84.4.1 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 84.3.1 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).
$\square$

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