Augmented chain complex ($\cdots \to C_1\to C_0\stackrel{\epsilon }{\to }\mathbb{Z}\to 0$)

An augmented chain complex includes an augmentation $\epsilon :C_0\to \mathbb{Z}$ with $\epsilon d_1=0$. For a nonempty simplicial complex, the augmentation sends every vertex to $1$ and the augmented homology in degree zero is its reduced homology.