The Stacks project

83.21 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 83.19 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

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

by thinking of $U$ as a simiplical semi-representable object of $\mathcal{C}/X$ whose degree $n$ part is the singleton element $\{ U_ n/X\} $ and applying the constructions in Remark 83.16.4.

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!

Let us say that 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 $n \geq 1$.

The category $\mathcal{C}/X$ has all connected finite limits, hence the coskeleta used in the formulation above exist. Of course, we see that $U$ is a hypercovering of $X$ in $\mathcal{C}$ if and only if the simplicial semi-representable object $\{ U_ n\} $ is a hypercovering of $X$ in the sense of Section 83.19.

Lemma 83.21.1. 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 83.19.1. $\square$

Lemma 83.21.2. 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 83.19.2. $\square$

Lemma 83.21.3. 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 83.19.3. $\square$

Lemma 83.21.4. 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 83.19.4 $\square$

Lemma 83.21.5. 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 83.3.1 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 83.4.1 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 83.3.1 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). $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09X8. Beware of the difference between the letter 'O' and the digit '0'.