## Tag `017P`

Chapter 14: Simplicial Methods > Section 14.18: Splitting simplicial objects

Definition 14.18.1. Let $\mathcal{C}$ be a category which admits finite nonempty coproducts. We say a simplicial object $U$ of $\mathcal{C}$ is

splitif there exist subobjects $N(U_m)$ of $U_m$, $m \geq 0$ with the property that \begin{equation} \tag{14.18.1.1} \coprod\nolimits_{\varphi : [n] \to [m]\text{ surjective}} N(U_m) \longrightarrow U_n \end{equation} is an isomorphism for all $n \geq 0$. If $U$ is an $r$-truncated simplicial object of $\mathcal{C}$ then we say $U$ issplitif there exist subobjects $N(U_m)$ of $U_m$, $r \geq m \geq 0$ with the property that (14.18.1.1) is an isomorphism for $r \geq n \geq 0$.

The code snippet corresponding to this tag is a part of the file `simplicial.tex` and is located in lines 1755–1773 (see updates for more information).

```
\begin{definition}
\label{definition-split}
Let $\mathcal{C}$ be a category which admits finite nonempty coproducts.
We say a simplicial object $U$ of $\mathcal{C}$ is {\it split}
if there exist subobjects $N(U_m)$ of $U_m$, $m \geq 0$
with the property that
\begin{equation}
\label{equation-splitting}
\coprod\nolimits_{\varphi : [n] \to [m]\text{ surjective}}
N(U_m)
\longrightarrow
U_n
\end{equation}
is an isomorphism for all $n \geq 0$. If $U$ is an $r$-truncated
simplicial object of $\mathcal{C}$ then we say $U$ is {\it split}
if there exist subobjects $N(U_m)$ of $U_m$, $r \geq m \geq 0$
with the property that (\ref{equation-splitting})
is an isomorphism for $r \geq n \geq 0$.
\end{definition}
```

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