Homology of a chain complex

Summary

The homology of a chain complex $(C_\bullet, d)$ is the sequence of objects $H_n(C) = \ker d_n / \im d_{n+1}$. It measures the failure of the complex to be exact: $H_n(C) = 0$ exactly when the complex is exact at $C_n$. Each chain map induces maps on homology, so homology is a functor. The homology of the singular chain complex of a space is its singular homology. The same construction gives simplicial homology, group homology and derived functors.