Definition 25.3.1. Let $\mathcal{C}$ be a site. Let $f = (\alpha , f_ i) : \{ U_ i\} _{i \in I} \to \{ V_ j\} _{j \in J}$ be a morphism in the category $\text{SR}(\mathcal{C})$. We say that $f$ is a covering if for every $j \in J$ the family of morphisms $\{ U_ i \to V_ j\} _{i \in I, \alpha (i) = j}$ is a covering for the site $\mathcal{C}$. Let $X$ be an object of $\mathcal{C}$. A morphism $K \to L$ in $\text{SR}(\mathcal{C}, X)$ is a covering if its image in $\text{SR}(\mathcal{C})$ is a covering.
25.3 Hypercoverings
If we assume our category is a site, then we can make the following definition.
Lemma 25.3.2. Let $\mathcal{C}$ be a site.
A composition of coverings in $\text{SR}(\mathcal{C})$ is a covering.
If $K \to L$ is a covering in $\text{SR}(\mathcal{C})$ and $L' \to L$ is a morphism, then $L' \times _ L K$ exists and $L' \times _ L K \to L'$ is a covering.
If $\mathcal{C}$ has products of pairs, and $A \to B$ and $K \to L$ are coverings in $\text{SR}(\mathcal{C})$, then $A \times K \to B \times L$ is a covering.
Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Then (1) and (2) holds for $\text{SR}(\mathcal{C}, X)$ and (3) holds if $\mathcal{C}$ has fibre products.
Proof. Part (1) is immediate from the axioms of a site. Part (2) follows by the construction of fibre products in $\text{SR}(\mathcal{C})$ in the proof of Lemma 25.2.3 and the requirement that the morphisms in a covering of $\mathcal{C}$ are representable. Part (3) follows by thinking of $A \times K \to B \times L$ as the composition $A \times K \to B \times K \to B \times L$ and hence a composition of basechanges of coverings. The final statement follows because $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$. $\square$
By Lemma 25.2.3 and Simplicial, Lemma 14.19.2 the coskeleton of a truncated simplicial object of $\text{SR}(\mathcal{C}, X)$ exists if $\mathcal{C}$ has fibre products. Hence the following definition makes sense.
Definition 25.3.3. Let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has fibre products. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be an object of $\mathcal{C}$. A hypercovering of $X$ is a simplicial object $K$ of $\text{SR}(\mathcal{C}, X)$ such that
The object $K_0$ is a covering of $X$ for the site $\mathcal{C}$.
For every $n \geq 0$ the canonical morphism
is a covering in the sense defined above.
\[ K_{n + 1} \longrightarrow (\text{cosk}_ n \text{sk}_ n K)_{n + 1} \]
Condition (1) makes sense since each object of $\text{SR}(\mathcal{C}, X)$ is after all a family of morphisms with target $X$. It could also be formulated as saying that the morphism of $K_0$ to the final object of $\text{SR}(\mathcal{C}, X)$ is a covering.
Example 25.3.4 (Čech hypercoverings). Let $\mathcal{C}$ be a site with fibre products. Let $\{ U_ i \to X\} _{i \in I}$ be a covering of $\mathcal{C}$. Set $K_0 = \{ U_ i \to X\} _{i \in I}$. Then $K_0$ is a $0$-truncated simplicial object of $\text{SR}(\mathcal{C}, X)$. Hence we may form Clearly $K$ passes condition (1) of Definition 25.3.3. Since all the morphisms $K_{n + 1} \to (\text{cosk}_ n \text{sk}_ n K)_{n + 1}$ are isomorphisms by Simplicial, Lemma 14.19.10 it also passes condition (2). Note that the terms $K_ n$ are the usual A hypercovering of $X$ of this form is called a Čech hypercovering of $X$.
\[ K = \text{cosk}_0 K_0. \]
\[ K_ n = \{ U_{i_0} \times _ X U_{i_1} \times _ X \ldots \times _ X U_{i_ n} \to X \} _{(i_0, i_1, \ldots , i_ n) \in I^{n + 1}} \]
Example 25.3.5 (Hypercovering by a simplicial object of the site). Let $\mathcal{C}$ be a site with fibre products. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $U$ be a simplicial object of $\mathcal{C}$. As usual we denote $U_ n = U([n])$. Finally, assume given an augmentation In this situation we can consider the simplicial object $K$ of $\text{SR}(\mathcal{C}, X)$ with terms $K_ n = \{ U_ n \to X\} $. Then $K$ is a hypercovering of $X$ in the sense of Definition 25.3.3 if and only if the following three conditions1 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 $n \geq 1$.
\[ a : U \to X \]
We omit the straightforward verification.
Example 25.3.6 (Čech hypercovering associated to a cover). Let $\mathcal{C}$ be a site with fibre products. Let $U \to X$ be a morphism of $\mathcal{C}$ such that $\{ U \to X\} $ is a covering of $\mathcal{C}$2. Consider the simplical object $K$ of $\text{SR}(\mathcal{C}, X)$ with terms Then $K$ is a hypercovering of $X$. This example is a special case of both Example 25.3.4 and of Example 25.3.5.
\[ K_ n = \{ U \times _ X U \times _ X \ldots \times _ X U \to X\} \quad (n + 1 \text{ factors}) \]
Lemma 25.3.7. Let $\mathcal{C}$ be a site with fibre products. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be an object of $\mathcal{C}$. The collection of all hypercoverings of $X$ forms a set.
Proof. Since $\mathcal{C}$ is a site, the set of all coverings of $X$ forms a set. Thus we see that the collection of possible $K_0$ forms a set. Suppose we have shown that the collection of all possible $K_0, \ldots , K_ n$ form a set. Then it is enough to show that given $K_0, \ldots , K_ n$ the collection of all possible $K_{n + 1}$ forms a set. And this is clearly true since we have to choose $K_{n + 1}$ among all possible coverings of $(\text{cosk}_ n \text{sk}_ n K)_{n + 1}$. $\square$
Remark 25.3.8. The lemma does not just say that there is a cofinal system of choices of hypercoverings that is a set, but that really the hypercoverings form a set.
The category of presheaves on $\mathcal{C}$ has finite (co)limits. Hence the functors $\text{cosk}_ n$ exists for presheaves of sets.
Lemma 25.3.9. Let $\mathcal{C}$ be a site with fibre products. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Consider the simplicial object $F(K)$ of $\textit{PSh}(\mathcal{C})$, endowed with its augmentation to the constant simplicial presheaf $h_ X$.
The morphism of presheaves $F(K)_0 \to h_ X$ becomes a surjection after sheafification.
The morphism
becomes a surjection after sheafification.
For every $n \geq 1$ the morphism
turns into a surjection after sheafification.
\[ (d^1_0, d^1_1) : F(K)_1 \longrightarrow F(K)_0 \times _{h_ X} F(K)_0 \]
\[ F(K)_{n + 1} \longrightarrow (\text{cosk}_ n \text{sk}_ n F(K))_{n + 1} \]
Proof. We will use the fact that if $\{ U_ i \to U\} _{i \in I}$ is a covering of the site $\mathcal{C}$, then the morphism
\[ \amalg _{i \in I} h_{U_ i} \to h_ U \]
becomes surjective after sheafification, see Sites, Lemma 7.12.4. Thus the first assertion follows immediately.
For the second assertion, note that according to Simplicial, Example 14.19.1 the simplicial object $\text{cosk}_0 \text{sk}_0 K$ has terms $K_0 \times \ldots \times K_0$. Thus according to the definition of a hypercovering we see that $(d^1_0, d^1_1) : K_1 \to K_0 \times K_0$ is a covering. Hence (2) follows from the claim above and the fact that $F$ transforms products into fibred products over $h_ X$.
For the third, we claim that $\text{cosk}_ n \text{sk}_ n F(K) = F(\text{cosk}_ n \text{sk}_ n K)$ for $n \geq 1$. To prove this, denote temporarily $F'$ the functor $\text{SR}(\mathcal{C}, X) \to \textit{PSh}(\mathcal{C})/h_ X$. By Lemma 25.2.3 the functor $F'$ commutes with finite limits. By our description of the $\text{cosk}_ n$ functor in Simplicial, Section 14.12 we see that $\text{cosk}_ n \text{sk}_ n F'(K) = F'(\text{cosk}_ n \text{sk}_ n K)$. Recall that the category used in the description of $(\text{cosk}_ n U)_ m$ in Simplicial, Lemma 14.19.2 is the category $(\Delta /[m])^{opp}_{\leq n}$. It is an amusing exercise to show that $(\Delta /[m])_{\leq n}$ is a connected category (see Categories, Definition 4.16.1) as soon as $n \geq 1$. Hence, Categories, Lemma 4.16.2 shows that $\text{cosk}_ n \text{sk}_ n F'(K) = \text{cosk}_ n \text{sk}_ n F(K)$. Whence the claim. Property (2) follows from this, because now we see that the morphism in (2) is the result of applying the functor $F$ to a covering as in Definition 25.3.1, and the result follows from the first fact mentioned in this proof. $\square$
[1] As $\mathcal{C}$ has fibre products, the category $\mathcal{C}/X$ has all finite limits. Hence the required coskeleta exist by Simplicial, Lemma 14.19.2.
[2] A morphism of $\mathcal{C}$ with this property is sometimes called a “cover”.
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