nLab
augmentation

Idea

An augmentation of a simplicial set or generally a simplicial object $S_{•}$ is a homomorphism of simplicial objects to a simplicial onbject constant (discrete) on an object $A$:

$$ϵ:S_{•}\to A.$$\epsilon \colon S_\bullet \to A \,.

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

$$\begin{array}{c}\cdots S_2\overrightarrow{\overrightarrow{\to }}{S}_1\overrightarrow{\to }{S}_0\stackrel{{ϵ}_0}{\to }A\end{array}$$\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 $V_{•}$ (in non-negative degree) is a chain map

$$ϵ:V_{•}\to A.$$\epsilon \colon V_\bullet \to A \,.

If $V_{•}$ and $A$ are equipped with algebra-structure ($V$ might be an augmented algebra over $A$), then the kernel of the augmentation map is called the augmentation ideal.

Related concepts

• reduced homology

.