Example 25.3.4 (01G6): Čech hypercoverings

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

\[ K = \text{cosk}_0 K_0. \]

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

\[ K_ n = \{ U_{i_0} \times _ X U_{i_1} \times _ X \ldots \times _ X U_{i_ n} \to X \} _{(i_0, i_1, \ldots , i_ n) \in I^{n + 1}} \]

A hypercovering of $X$ of this form is called a Čech hypercovering of $X$.

Comments (2)

Comment #1025 by correction_bot on September 09, 2014 at 07:34

To see that $K_{n + 1} \to (\text{cosk}_n \text{sk}_n K)_{n + 1}$ is an isomorphism, perhaps it is useful to reference Lemma 14.17.11(3) (tag 018B), which shows $K \to \text{cosk}_n \text{sk}_n K$ is an isomorphism.

Comment #1031 by Johan on September 09, 2014 at 13:22

Many thanks for all the comments. I have incorporated all of your suggestions. The changes can be found in this commit.

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