Singular simplices and singular chain groups with coefficients

Let $X$ be a topological space. For each integer $n\ge 0$, a singular $n$-simplex in $X$ is a continuous map $\sigma :{\mathrm{\Delta }}^n\to X$ from the standard topological simplex of The standard topological simplex and its affine face maps. Write $S_n(X)$ for the set of all singular $n$-simplices in $X$.

The singular chain group with integer coefficients is the free abelian group on $S_n(X)$: $C_n(X;\mathbb{Z}):={⨁}_{\sigma \in S_n(X)}\mathbb{Z}[\sigma ]\mathrm{.}$ Thus an element of $C_n(X;\mathbb{Z})$ is a finite formal sum ${\sum }_{k=1}^m{a}_k{\sigma }_k,a_k\in \mathbb{Z},{\sigma }_k\in S_n(X)\mathrm{.}$

If $G$ is an abelian group, the singular chain group with coefficients in $G$ is $C_n(X;G):=C_n(X;\mathbb{Z}){\otimes }_{\mathbb{Z}}G,$ using the tensor product of The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums. The page's convention is that coefficients are attached through this tensor-product construction, not by choosing a preferred $G$-basis of singular simplices.