---
tag: 99ZVSM254K
kind: example
name: Category of abelian groups
msc: ["18E10", 20K99]
instance_of: [Y9CZ3MTNF1]
uses: [082GHNPJ53, 0JYTG5K9Y4]
sources:
  - url: https://en.wikipedia.org/wiki/Category_of_abelian_groups
same_as:
  wikidata: Q1956793
  wikipedia: Category of abelian groups
---

## Example

The *category of abelian groups* $\mathbf{Ab}$ has the [[abelian groups|082GHNPJ53]] as objects and the [[group homomorphisms|0JYTG5K9Y4]] between them as morphisms, with composition of maps.
It is an [[abelian category|Y9CZ3MTNF1]].
Each set $\Hom(A, B)$ is an abelian group under pointwise addition $(f + g)(a) = f(a) + g(a)$, and composition is bilinear.
The trivial group is a [[zero object|AQET9YHH6F]], and $A \oplus B$ is both the product and the coproduct of $A$ and $B$.
Every homomorphism $f \colon A \to B$ has a [[kernel|S417YNEJ6M]] $\set{a \in A : f(a) = 0}$ and a [[cokernel|H8W38933NC]] $B / f(A)$.
An injective homomorphism $f$ is the kernel of the projection $B \to B/f(A)$.
A surjective homomorphism is the cokernel of the inclusion of its kernel.
So every monomorphism is a kernel and every epimorphism is a cokernel.

## Remarks

$\mathbf{Ab}$ is the model case of an abelian category: the [[Freyd–Mitchell embedding theorem|8V296XR2M3]] says that each small abelian category embeds fully and exactly into a category of modules.
It is also a full subcategory of the [[category of groups|2CX7RPPXA9]], but the coproducts differ: in $\mathbf{Ab}$ the coproduct of $\Z$ and $\Z$ is $\Z \oplus \Z$, not the free product.
The group $\Z$ is a [[generator|CK08JW32WB]], and $\Q/\Z$ is an [[injective object|5QRFTBF4PG]].
$\mathbf{Ab}$ is the same as the [[category of modules|JBSSXP4FFY]] over the ring $\Z$.

## Explanations

### L1 human

In the category of abelian groups, maps between two groups can themselves be added, because the groups are commutative.
This extra structure lets one talk about kernels, images and exact sequences, which is the starting point of homological algebra.
For example, multiplication by $2$ on $\Z$ is a morphism with kernel $0$ and cokernel $\Z/2\Z$.