augmentation

An **augmentation** of a simplicial set or generally a simplicial object $S_\bullet$ is a homomorphism of simplicial objects to a simplicial onbject 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**.

.

Revised on June 19, 2013 20:19:36
by Urs Schreiber
(82.169.65.155)