abelian group in nLab

Contents

Contents

Definition

An abelian group (named after Niels Henrik Abel) is a group AA where the multiplication satisfies the commutative law: for all elements x,y∈Ax, 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×A→A(-)-(-):A \times A \to A called subtraction such that

• for all a∈Aa \in A, a−a=0a - a = 0

• for all a∈Aa \in A, 0−(0−a)=a0 - (0 - a) = a

• for all a∈Aa \in A and b∈Ab \in A, a−(0−b)=b−(0−a)a - (0 - b) = b - (0 - a)

• for all a∈Aa \in A, b∈Ab \in A, and c∈Ac \in A, a−(b−c)=(a−(0−c))−ba - (b - c) = (a - (0 - c)) - b

For every element a∈Aa \in A, the inverse element is defined as −a≔0−a-a \coloneqq 0 - a and addition is defined as a+b≔a−(−b)a + b \coloneqq a - (-b).

Addition is commutative:

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

and associative

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

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

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

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

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

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

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

(a+b)+c=a−(0−(b−(0−c)))(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−(0−a)=a0 + a = 0 - (0 - a) = a

and right identities

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

and has left inverses

−a+a=(0−a)−(0−a)=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 • ⇓ a∈A • ↘↗ 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 a∈Aa \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.

• commutative magma

• commutative invertible semigroup

• 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

• super abelian group, super module

• quadratic abelian group

• tensor ring, Clifford ring

• Ab

• sheaf of abelian groups

• Ab-enriched category, abelian category

• abelian ∞-group

  • symmetric 2-group

  • symmetric 3-group

• halving group

• symmetric midpoint group

• abelian groupal setoid

• abelian Chern-Simons theory

References

Textbook account:

• László Fuchs, Abelian Groups, Springer (2015) [doi:10.1007/978-3-319-19422-6]

Formalization of abelian groups in univalent foundations of mathematics (homotopy type theory with the univalence axiom):

• Univalent Foundations Project, Section 6.11 of: Homotopy Type Theory – Univalent Foundations of Mathematics (2013)

• Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson, Section 4.12 of: Symmetry (2021)