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
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.
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:
and associative
and has left identities
and right identities
and has left inverses
and right identities
Thus, these axioms form an abelian group.
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 $A$.
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.
A monoid in Ab with its standard monoidal category structure, equivalently a (pointed) Ab-enriched category with a single object, is a ring.
Generalizations of the notion of abelian group in higher category theory include
notably abelian simplicial groups
and spectra.
An abelian group may also be seen as a discrete compact closed category.
Textbook account:
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)
Last revised on February 3, 2023 at 17:08:23. See the history of this page for a list of all contributions to it.