An augmentation of a simplicial set or generally a simplicial object S •S_\bullet is a homomorphism of simplicial objects to a simplicial object constant (discrete) on an object AA:
Equivalently this is an augmented simplicial object, namely a diagram of the form
(showing here only the face maps).
Under the Dold-Kan correspondence this yields:
The augmentation of a chain complex V •V_\bullet (in non-negative degree) is a chain map
If V •V_\bullet and AA are equipped with algebra-structure (VV might be an augmented algebra over AA), then the kernel of the augmentation map is called the augmentation ideal.
augmented algebra, augmented A-infinity algebra