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 $\mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(A, U)$. Each of these are canonically split by Lemma 14.18.5. Moreover,

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

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

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