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abelian group

Definition

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

$$x y = y x \,.$$

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 $\mathbf{Z}$. That is, Ab = $\mathbf{Z}$-Mod.

Properties

In homotopy theory

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

$$\mathbf{B}^2 A = \left\{ \array{ & \nearrow \searrow^{\mathrlap{Id}} \\ \bullet &\Downarrow^{a \in A}& \bullet \\ & \searrow \nearrow_{\mathrlap{Id}} } \right\} \,.$$

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

• abelian group objects in an (∞,1)-category

• notably abelian simplicial groups

• and spectra.

An abelian group may also be seen as a discrete compact closed category.

Related entries

• commutative magma

• tensor product of abelian groups, direct sum of abelian groups

• free abelian group, finite abelian group

• theory of abelian groups

• abelianization

• anabelian group

• module, ring

• commutative ring, commutative algebra

• Ab

• sheaf of abelian groups

• Ab-enriched category, abelian category

• abelian ∞-group