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
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.
Comments (0)