group theory

# Contents

## Definition

An abelian group (named after Nils Abel) is a group $A$ where the multiplication satisfies the commutative law: for all elements $x,y\in A$ we have

$xy=yx\phantom{\rule{thinmathspace}{0ex}}.$x y = y x \,.

The category with abelian groups as objects and group homomorphisms as morphisms is called Ab.

## Properties

### In homtopy theory

From the nPOV, just as a group $G$ may be thought of as a (pointed) groupoid $BG$ with a single object – as discussed at delooping – an abelian group $A$ may be understood as a (pointed) 2-groupoid ${B}^{2}A$ with a single object and a single morphism: the delooping of the delooping of $A$.

${B}^{2}A=\left\{\begin{array}{cc}& ↗{↘}^{\mathrm{Id}}\\ •& {⇓}^{a\in A}& •\\ & ↘{↗}_{\mathrm{Id}}\end{array}\right\}\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{B}^2 A = \left\{ \array{ & \nearrow \searrow^{\mathrlap{Id}} \\ \bullet &\Downarrow^{a \in A}& \bullet \\ & \searrow \nearrow_{\mathrlap{Id}} } \right\} \,.

The exchange law for the composition of 2-morphisms in a 2-category forces the product on the $a\in A$ here to be commutative. This reasoning is known as the Eckmann-Hilton argument and is the same as the reasoning that finds that the second homotopy group of a space has to be abelian.

So the identitfication of abelian groups with one-object, one-morphism 2-groupoids may also be thought of as an identification with 2-truncated and 2-connected homotopy types.

### Relation to other concepts

A monoid in Ab with its standard monoidal category structure, equivalently a (pointed) Ab-enriched category with a single object, is a ring.

## Generalizations

Generalizations of the notion of abelian group in higher category theory include

Revised on March 31, 2013 12:07:24 by Urs Schreiber (82.113.99.216)