The Stacks project

Lemma 25.4.3. Let $\mathcal{C}$ be a site. Let $f : L \to K$ be a morphism of simplicial objects of $\textit{PSh}(\mathcal{C})$. Let $n \geq 0$ be an integer. Assume that

  1. For $i < n$ the morphism $L_ i \to K_ i$ is an isomorphism.

  2. The morphism $L_ n \to K_ n$ is surjective after sheafification.

  3. The canonical map $L \to \text{cosk}_ n \text{sk}_ n L$ is an isomorphism.

  4. The canonical map $K \to \text{cosk}_ n \text{sk}_ n K$ is an isomorphism.

Then $H_ i(f) : H_ i(L) \to H_ i(K)$ is an isomorphism.

Proof. This proof is exactly the same as the proof of Lemma 25.4.2 above. Namely, we first let $K_ n' \subset K_ n$ be the sub presheaf which is the image of the map $L_ n \to K_ n$. Assumption (2) means that the sheafification of $K_ n'$ is equal to the sheafification of $K_ n$. Moreover, since $L_ i = K_ i$ for all $i < n$ we see that get an $n$-truncated simplicial presheaf $U$ by taking $U_0 = L_0 = K_0, \ldots , U_{n - 1} = L_{n - 1} = K_{n - 1}, U_ n = K'_ n$. Denote $K' = \text{cosk}_ n U$, a simplicial presheaf. Because we can construct $K'_ m$ as a finite limit, and since sheafification is exact, we see that $(K'_ m)^\# = K_ m$. In other words, $(K')^\# = K^\# $. We conclude, by exactness of sheafification once more, that $H_ i(K) = H_ i(K')$. Thus it suffices to prove the lemma for the morphism $L \to K'$, in other words, we may assume that $L_ n \to K_ n$ is a surjective morphism of presheaves!

In this case, for any object $U$ of $\mathcal{C}$ we see that the morphism of simplicial sets

\[ L(U) \longrightarrow K(U) \]

satisfies all the assumptions of Simplicial, Lemma 14.32.1. Hence it is a trivial Kan fibration. In particular it is a homotopy equivalence (Simplicial, Lemma 14.30.8). Thus

\[ \mathbf{Z}_ L(U) \longrightarrow \mathbf{Z}_ K(U) \]

is a homotopy equivalence too. This for all $U$. The result follows. $\square$

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Your email address will not be published.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 01GD. Beware of the difference between the letter 'O' and the digit '0'.