nLab 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.

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 $(A, 0)$ with a binary operation $(-)-(-):A \times A \to A$ called subtraction such that

• for all $a \in A$, $a - a = 0$

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

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

• for all $a \in A$, $b \in A$, and $c \in A$, $a - (b - c) = (a - (0 - c)) - b$

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

Addition is commutative:

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

and associative

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

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

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

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

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

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

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

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

$$(a + b) + c = a + (b + c)$$

and has left identities

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

and right identities

$$a + 0 = 0 + a = a$$

and has left inverses

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

and right identities

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

Thus, these axioms form an abelian group.

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

• 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

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)