The Stacks project

Lemma 85.19.1. Let $\mathcal{C}$ be a site with fibre products and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $K$ be a hypercovering of $X$. Then

  1. $a^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X) \to \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of sets,

  2. $a^{-1} : \textit{Ab}(\mathcal{C}/X) \to \textit{Ab}((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups.

In both cases $a_*$ provides the quasi-inverse functor.

Proof. Via Remarks 85.15.5 and 85.16.4 and the discussion in the introduction to this section this follows from Lemma 85.17.1. $\square$

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Your email address will not be published.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 0D8K. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 85.19.1 as pdf