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 split if there exist subobjects $N(U_ m)$ of $U_ m$, $m \geq 0$ with the property that

14.18.1.1
$$\label{simplicial-equation-splitting} \coprod \nolimits _{\varphi : [n] \to [m]\text{ surjective}} N(U_ m) \longrightarrow U_ n$$

is an isomorphism for all $n \geq 0$. If $U$ is an $r$-truncated simplicial object of $\mathcal{C}$ then we say $U$ is split if 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 $r \geq n \geq 0$.

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