Problem. Given a simplicial set $X$, we can associate to it a chain complex defined by $$C_n=\mathbb{Z}[X_n]$$(the free abelian group on the set $X_n$) with differentials $$d=\sum_i(-1)^id_i:C_n\to C_{n-1},$$ the alternating sum of the face maps. Is the homology of this chain complex the same as the singular homology of the geometric realization, i.e. do we have $$H(C_*)\cong H\left(|X|;\mathbb{Z}\right)\quad ?$$
Attempt. There is a natural map on the level of chain complexes given by
$$\begin{align}C_n \longrightarrow C_n^{sing}\left(|X|\right)\\ x_n\mapsto\left(t\mapsto[t,x_n]\right)\end{align}$$
and I wonder if this induces an isomorphism on homology.
Alternatively, $|X|$ is a CW complex with one $n$-cell $e_n^i$ for each nondegenerate $n$-simplex $x_n^i$, so we can look at the chain map
$$\begin{align} C_n^{cell}\left( |X| \right) \longrightarrow C_n\\ e_n^i\mapsto x_n^i \end{align}$$
Is this a homology isomorphism?