An augmentation of a simplicial set or generally a simplicial object S S_\bullet is a homomorphism of simplicial objects to a simplicial onbject constant (discrete) on an object AA:

ϵ:S A. \epsilon \colon S_\bullet \to A \,.

Equivalently this is an augmented simplicial object, namely a diagram of the form

S 2S 1S 0ϵ 0A \array{ \cdots S_2 \stackrel{\to}{\stackrel{\to}{\to}} S_1 \stackrel{\to}{\to} S_0 \stackrel{\epsilon_0}{\to} A }

(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

ϵ:V A. \epsilon \colon V_\bullet \to A \,.

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.


Revised on June 19, 2013 20:19:36 by Urs Schreiber (