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Tag 017V

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

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 $$ N(U_m) = \bigcap\nolimits_{i = 0}^{m - 1} \mathop{\rm Ker}(d^m_i). $$ 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 $\mathop{\rm Mor}\nolimits_\mathcal{A}(A, U)$. Each of these are canonically split by Lemma 14.18.5. Moreover, $$ N(\mathop{\rm Mor}\nolimits_\mathcal{A}(A, U_m)) = \bigcap\nolimits_{i = 0}^{m - 1} \mathop{\rm Ker}(d^m_i) = \mathop{\rm Mor}\nolimits_\mathcal{A}(A, N(U_m)). $$ Hence we see that the morphism (14.18.1.1) becomes an isomorphism after applying the functor $\mathop{\rm Mor}\nolimits_\mathcal{A}(A, -)$ for any object of $\mathcal{A}$. Hence it is an isomorphism by the Yoneda lemma. $\square$

    The code snippet corresponding to this tag is a part of the file simplicial.tex and is located in lines 2017–2028 (see updates for more information).

    \begin{lemma}
    \label{lemma-splitting-abelian-category}
    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
    $$
    N(U_m) = \bigcap\nolimits_{i = 0}^{m - 1} \Ker(d^m_i).
    $$
    Moreover, this splitting is functorial on the category of
    simplicial objects of $\mathcal{A}$.
    \end{lemma}
    
    \begin{proof}
    For any object $A$ of $\mathcal{A}$ we obtain
    a simplicial abelian group $\Mor_\mathcal{A}(A, U)$.
    Each of these are canonically split by Lemma
    \ref{lemma-splitting-simplicial-groups}. Moreover,
    $$
    N(\Mor_\mathcal{A}(A, U_m)) =
    \bigcap\nolimits_{i = 0}^{m - 1} \Ker(d^m_i) =
    \Mor_\mathcal{A}(A, N(U_m)).
    $$
    Hence we see that the morphism (\ref{equation-splitting})
    becomes an isomorphism after applying the functor
    $\Mor_\mathcal{A}(A, -)$ for any object of $\mathcal{A}$.
    Hence it is an isomorphism by the Yoneda lemma.
    \end{proof}

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