# Contents

## 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 $A$:

$ϵ:{S}_{•}\to A\phantom{\rule{thinmathspace}{0ex}}.$\epsilon \colon S_\bullet \to A \,.

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

$\begin{array}{c}\cdots {S}_{2}\stackrel{\to }{\stackrel{\to }{\to }}{S}_{1}\stackrel{\to }{\to }{S}_{0}\stackrel{{ϵ}_{0}}{\to }A\end{array}$\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

$ϵ:{V}_{•}\to A\phantom{\rule{thinmathspace}{0ex}}.$\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.

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