Augmented singular chain complex

For coefficients over the integers
The augmented singular chain complex is a slight variant of the singular chain complex where we define $$C_{-1}(X)$$ to equal $$\mathbb{Z}$$ and the boundary map $$\partial_0: C_0(X) \to C_{-1}(X)$$ as the sum of coefficients function, also called the augmentation map. Details below: