Augmented singular complex

Definition

The augmented singular complex is the singular complex with the following change: the $-1^{th}$ chain group (which is zero in the singular complex) is now the group of integers, and the map from the zeroth chain group to this is the map which takes a chain and simply sends it to the sum of coefficients of all the points.

We can motivate this by thinking of the empty set as the standard $(-1)$-simplex.

This is in fact a particular case of the general notion of augmentation of nonnegative complex.