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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 split if 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 $U$ 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 $r \geq n \geq 0$.

    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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