= Simplicial chain group
{title2=$C_n(K;A)$}
For an <abstract simplicial complex> $K$, the integer <simplicial chain group> $C_n(K;\mathbb Z)$ is the <free abelian group> on its oriented $n$-<simplexes>, with reversal of orientation changing sign. Thus a chain is a finite formal integer sum of oriented simplices. After choosing one orientation for each simplex, those chosen simplices form a <basis> of this <abelian group>.
For a filled <triangle> with vertices $0,1,2$, $C_0\cong\mathbb Z^3$, $C_1\cong\mathbb Z^3$ and $C_2\cong\mathbb Z$. Its oriented two-simplex has boundary
$$
\partial[0,1,2]=[1,2]-[0,2]+[0,1].
$$
Taking another boundary gives $( [2]-[1])-([2]-[0])+([1]-[0])=0$. This cancellation makes the <simplicial chain complex> possible: boundaries are cycles, and <simplicial homology> measures cycles modulo boundaries. With coefficients in an <abelian group> $A$, use $C_n(K;\mathbb Z)\otimes_{\mathbb Z}A$.<Algebraic Topology by Allen Hatcher>[section 2.1, simplicial homology and boundary maps]{ref}
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