nLab augmentation

Idea

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

$$\epsilon \colon S_\bullet \to A \,.$$

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

$$\array{ \cdots S_2 \stackrel{\to}{\stackrel{\to}{\to}} S_1 \stackrel{\to}{\to} S_0 \stackrel{\epsilon_0}{\to} A }$$

(showing here only the face maps).

Under the Dold-Kan correspondence this yields:

The augmentation of a chain complex $V_\bullet$ (in non-negative degree) is a chain map

$$\epsilon \colon V_\bullet \to A \,.$$

If $V_\bullet$ 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

• augmented algebra, augmented A-infinity algebra

• augmentation ideal

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