aka group of n-chains

Definition. The Free Abelian group generated by the simplices of a Delta-complex on $X$, denoted $\Delta_n(X) := \mathbb{Z} \langle\{e^n_\alpha\}\rangle$. An element of the group is called an n-chain

One can define it as the free Abelian group generated by other sets of simplices, like all singular simplices, for Singular homology.

See video