The Stacks project

Remark 25.8.4. A useful special case of Lemmas 25.8.2 and 25.8.3 is the following. Suppose we have a category $\mathcal{C}$ having fibre products. Let $P \subset \text{Arrows}(\mathcal{C})$ be a subset stable under base change, stable under composition, and containing all isomorphisms. Then one says a $P$-hypercovering is an augmentation $a : U \to X$ from a simplicial object of $\mathcal{C}$ such that

  1. $U_0 \to X$ is in $P$,

  2. $U_1 \to U_0 \times _ X U_0$ is in $P$,

  3. $U_{n + 1} \to (\text{cosk}_ n\text{sk}_ n U)_{n + 1}$ is in $P$ for $n \geq 1$.

The category $\mathcal{C}/X$ has all finite limits, hence the coskeleta used in the formulation above exist (see Categories, Lemma 4.18.4). Then we claim that the morphisms $U_ n \to X$ and $d^ n_ i : U_ n \to U_{n - 1}$ are in $P$. This follows from the aforementioned lemmas by turning $\mathcal{C}$ into a site whose coverings are $\{ f : V \to U\} $ with $f \in P$ and taking $K$ given by $K_ n = \{ U_ n \to X\} $.

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Your email address will not be published.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 0DER. Beware of the difference between the letter 'O' and the digit '0'.