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.

With subtraction and unit only

This definition of abelian group is based upon Toby Bartels‘s definition of an associative quasigroup:

An abelian group is a pointed set (A,0)(A, 0) with a binary operation ()():A×AA(-)-(-):A \times A \to A called subtraction such that

  • for all aAa \in A, aa=0a - a = 0

  • for all aAa \in A, 0(0a)=a0 - (0 - a) = a

  • for all aAa \in A and bAb \in A, a(0b)=b(0a)a - (0 - b) = b - (0 - a)

  • for all aAa \in A, bAb \in A, and cAc \in A, a(bc)=(a(0c))ba - (b - c) = (a - (0 - c)) - b

For every element aAa \in A, the inverse element is defined as a0a-a \coloneqq 0 - a and addition is defined as a+ba(b)a + b \coloneqq a - (-b).

Addition is commutative:

a+b=a(0b)=b(0a)=b+aa + b = a - (0 - b) = b - (0 - a) = b + a

and associative

(a+b)+c=(a(0b))(0c)(a + b) + c = (a - (0 - b)) - (0 - c)
(a+b)+c=(b(0a))(0c)(a + b) + c = (b - (0 - a)) - (0 - c)
(a+b)+c=b((0c)a)(a + b) + c = b - ((0 - c) - a)
(a+b)+c=b((0c)(0(0a)))(a + b) + c = b - ((0 - c) - (0 - (0 - a)))
(a+b)+c=b((0a)(0(0c)))(a + b) + c = b - ((0 - a) - (0 - (0 - c)))
(a+b)+c=b((0a)c)(a + b) + c = b - ((0 - a) - c)
(a+b)+c=(b(0c))(0a)(a + b) + c = (b - (0 - c)) - (0 - a)
(a+b)+c=a(0(b(0c)))(a + b) + c = a - (0 - (b - (0 - c)))
(a+b)+c=a+(b+c)(a + b) + c = a + (b + c)

and has left identities

0+a=0(0a)=a0 + a = 0 - (0 - a) = a

and right identities

a+0=0+a=aa + 0 = 0 + a = a

and has left inverses

a+a=(0a)(0a)=0-a + a = (0 - a) - (0 - a) = 0

and right identities

a+(a)=a+a=0a + (-a) = -a + a = 0

Thus, these axioms form an abelian group.

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

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

References

For abelian groups in the context of homotopy type theory:

Last revised on June 17, 2022 at 23:25:26. See the history of this page for a list of all contributions to it.