The Stacks project

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

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

  2. 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

\[ 0 \to K(A, k) \to E \to K(A, k + 1) \to 0 \]

which is term by term split exact.

Proof. The maps $K(A, k)_ n \to E_ n$ resp. $E_ n \to K(A, k + 1)_ n$ are given by the inclusion of direct sums, resp. projection of direct sums which is obvious from the inclusions of index sets. It is clear that these are maps of simplicial objects. $\square$

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