Lemma 14.18.6. Let $\mathcal{A}$ be an abelian category. Let $U$ be a simplicial object in $\mathcal{A}$. Then $U$ has a splitting obtained by taking $N(U_0) = U_0$ and for $m \geq 1$ taking

$N(U_ m) = \bigcap \nolimits _{i = 0}^{m - 1} \mathop{\mathrm{Ker}}(d^ m_ i).$

Moreover, this splitting is functorial on the category of simplicial objects of $\mathcal{A}$.

Proof. For any object $A$ of $\mathcal{A}$ we obtain a simplicial abelian group $\mathop{Mor}\nolimits _\mathcal {A}(A, U)$. Each of these are canonically split by Lemma 14.18.5. Moreover,

$N(\mathop{Mor}\nolimits _\mathcal {A}(A, U_ m)) = \bigcap \nolimits _{i = 0}^{m - 1} \mathop{\mathrm{Ker}}(d^ m_ i) = \mathop{Mor}\nolimits _\mathcal {A}(A, N(U_ m)).$

Hence we see that the morphism (14.18.1.1) becomes an isomorphism after applying the functor $\mathop{Mor}\nolimits _\mathcal {A}(A, -)$ for any object of $\mathcal{A}$. Hence it is an isomorphism by the Yoneda lemma. $\square$

## Comments (0)

There are also:

• 4 comment(s) on Section 14.18: Splitting simplicial objects

## 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Your email address will not be published.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 017V. Beware of the difference between the letter 'O' and the digit '0'.