nLab
abelian group

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Definition

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

xy=yx. x y = y x \,.

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

Every abelian group has the canonical structure of a module over the commutative ring Z\mathbf{Z}. That is, Ab = Z\mathbf{Z}-Mod.

Properties

In homotopy theory

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

B 2A={ Id aA Id}. \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 aAa \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 October 29, 2014 11:21:58 by Anonymous Coward (89.204.137.244)