Lemma 14.22.4. Let $\mathcal{A}$ be an abelian category. Let $A$ be an object of $\mathcal{A}$ and let $k$ be an integer $\geq 0$. Consider the simplicial object $E$ defined by the following rules

$E_ n = \bigoplus _\alpha A$, where the sum is over $\alpha : [n] \to [k + 1]$ whose image is either $[k]$ or $[k + 1]$.

Given $\varphi : [m] \to [n]$ the map $E_ n \to E_ m$ maps the summand corresponding to $\alpha $ via $\text{id}_ A$ to the summand corresponding to $\alpha \circ \varphi $, provided $\mathop{\mathrm{Im}}(\alpha \circ \varphi )$ is equal to $[k]$ or $[k + 1]$.

Then there exists a short exact sequence

which is term by term split exact.

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