Lemma 25.12.3. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Assume that any object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$. Then there is a hypercovering $K$ such that $K_ n = \{ U_ i\} _{i \in I_ n}$ with $U_ i \in \mathcal{B}$ for all $i \in I_ n$.

Proof. This proof is almost the same as the proof of Lemma 25.12.2. We will only explain the differences.

Let $n \geq 1$. Let us say that an $n$-truncated $\mathcal{B}$-hypercovering is given by an $n$-truncated simplicial object $K$ of $\text{SR}(\mathcal{C})$ such that for $i \in I_ a$, $0 \leq a \leq n$ we have $U_{a, i} \in \mathcal{B}$ and such that

1. $F(K_0)^\# \to *$ is surjective,

2. $F(K_1)^\# \to F(K_0)^\# \times F(K_0)^\#$ is surjective,

3. $F(K_{a + 1})^\# \to F((\text{cosk}_ a \text{sk}_ a K)_{a + 1})^\#$ for $a = 1, \ldots , n - 1$ is surjective.

We first explicitly construct a split $1$-truncated $\mathcal{B}$-hypercovering.

Take $I_0 = \mathcal{B}$ and $K_0 = \{ U\} _{U \in \mathcal{B}}$. Then (1) holds by our assumption on $\mathcal{B}$. Set

$\Omega = \{ (U, V, W, a, b) \mid U, V, W \in \mathcal{B}, a : U \to V, b : U \to W\}$

Then we set $I_1 = I_0 \amalg \Omega$. For $i \in I_1$ we set $U_{1, i} = U_{0, i}$ if $i \in I_0$ and $U_{1, i} = U$ if $i = (U, V, W, a, b) \in \Omega$. The map $K(\sigma ^0_0) : K_0 \to K_1$ corresponds to the inclusion $\alpha (\sigma ^0_0) : I_0 \to I_1$ and the identity $f_{\sigma ^0_0, i} : U_{0, i} \to U_{1, i}$ on objects. The maps $K(\delta ^1_0), K(\delta ^1_1) : K_1 \to K_0$ correspond to the two maps $I_1 \to I_0$ which are the identity on $I_0 \subset I_1$ and map $(U, V, W, a, b) \in \Omega \subset I_1$ to $V$, resp. $W$. The corresponding morphisms $f_{\delta ^1_0, i}, f_{\delta ^1_1, i} : U_{1, i} \to U_{0, i}$ are the identity if $i \in I_0$ and $a, b$ in case $i = (U, V, W, a, b) \in \Omega$. The reason that (2) holds is that any section of $F(K_0)^\# \times F(K_0)^\#$ over an object $U$ of $\mathcal{C}$ comes, after replacing $U$ by the members of a covering, from a map $U \to F(K_0) \times F(K_0)$. This in turn means we have $V, W \in \mathcal{B}$ and two morphisms $U \to V$ and $U \to W$. Further replacing $U$ by the members of a covering we may assume $U \in \mathcal{B}$ as desired.

The lemma follows if we can prove that given a split $n$-truncated $\mathcal{B}$-hypercovering $K$ for $n \geq 1$ we can extend it to a split $(n + 1)$-truncated $\mathcal{B}$-hypercovering. Here the argument proceeds exactly as in the proof of Lemma 25.12.2. We omit the precise details, except for the following comments. First, we do not need assumption (2) in the proof of the current lemma as we do not need the morphism $K_{n + 1} \to (\text{cosk}_ n K)_{n + 1}$ to be covering; we only need it to induce a surjection on associated sheaves of sets which follows from Sites, Lemma 7.12.4. Second, the assumption that $\mathcal{C}$ has fibre products and equalizers guarantees that $\text{SR}(\mathcal{C})$ has fibre products and equalizers and $F$ commutes with these (Lemma 25.2.3). This suffices assure us the coskeleton functors used exist (see Simplicial, Remark 14.19.11 and Categories, Lemma 4.18.2). $\square$

Comment #5242 by on

Are such hypercoverings cofinal in the category of all hypercoverings $\text{HC}(\mathcal{C}, X)$?

Comment #5243 by on

@#5242: Yes, I believe this is true in the sense that for every hypercovering $K$, there is a morphism of hypercoverings $K' \to K$ such that $K'$ is as in Lemma 25.12.3. I think this should be straightforward to prove by modifying the proof of the lemma. But one needs to be careful in applying this because the category of hypercoverings isn't cofiltered! On the other hand, the category of hypercoverings is cofiltered in a homotopy sense, see Lemma 25.9.2. See also Remark 25.9.3 in case you need higher homotopies as well.

Comment #5244 by on

@#5243: Ah! Thanks for pointing out the homotopy issue. The "cofinality" sometimes might be useful. For example, if we work on the category $\mathcal{C}$ of pairs $(X,Z)$ over a trait ($Z$ a proper closed subscheme of proper flat variety $X$) with some topology, then by de Jong's alteration, the hypercoverings consisting of semi-stable pairs are "homotopically" cofinal in $\text{HC}(\mathcal{C}, (X,Z))$. By Verdier's hypercovering theorem (SGA4 V 7.4.1.4), for any presheaf $\mathcal{F}$ on $\mathcal{C}$, we have $H^i((X,Z), \mathcal{F}^{\sharp})=\operatorname{colim}_{K \in \text{HC}(\mathcal{C}, (X,Z))} \check{H}^i(K, \mathcal{F})$. Thus, we can take $K$ running through the hypercoverings consisting of semi-stable pairs, and the information of the presheaf on such pairs will dominate the cohomology. Maybe I will go through the details of the "cofinality" if I really need this cohomology computation someday.

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