The Stacks project

Remark 14.23.7. In the situation of Lemma 14.23.6 the subcomplex $D(U) \subset s(U)$ can also be defined as the subcomplex with terms

\[ D(U)_ n = \mathop{\mathrm{Im}}\left( \bigoplus \nolimits _{\varphi : [n] \to [m]\text{ surjective}, \ m < n} U_ m \xrightarrow {\bigoplus U(\varphi )} U_ n\right) \]

Namely, since $U_ m$ is the direct sum of the subobject $N(U_ m)$ and the images of $N(U_ k)$ for surjections $[m] \to [k]$ with $k < m$ this is clearly the same as the definition of $D(U)_ n$ given in the proof of Lemma 14.23.6. Thus we see that if $U$ is a simplicial abelian group, then elements of $D(U)_ n$ are exactly the sums of degenerate $n$-simplices.

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