# Category of abelian groups

In mathematics, the category **Ab** has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category:[1] indeed, every small abelian category can be embedded in **Ab**.[2]

## Properties

The zero object of **Ab** is the trivial group {0} which consists only of its neutral element.

The monomorphisms in **Ab** are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms.

**Ab** is a full subcategory of **Grp**, the category of *all* groups. The main difference between **Ab** and **Grp** is that the sum of two homomorphisms *f* and *g* between abelian groups is again a group homomorphism:

- (
*f*+*g*)(*x*+*y*) =*f*(*x*+*y*) +*g*(*x*+*y*) =*f*(*x*) +*f*(*y*) +*g*(*x*) +*g*(*y*) - =
*f*(*x*) +*g*(*x*) +*f*(*y*) +*g*(*y*) = (*f*+*g*)(*x*) + (*f*+*g*)(*y*)

The third equality requires the group to be abelian. This addition of morphism turns **Ab** into a preadditive category, and because the direct sum of finitely many abelian groups yields a biproduct, we indeed have an additive category.

In **Ab**, the notion of kernel in the category theory sense coincides with kernel in the algebraic sense, i.e. the categorical kernel of the morphism *f* : *A* → *B* is the subgroup *K* of *A* defined by *K* = {*x* ∈ *A* : *f*(*x*) = 0}, together with the inclusion homomorphism *i* : *K* → *A*. The same is true for cokernels; the cokernel of *f* is the quotient group *C* = *B* / *f*(*A*) together with the natural projection *p* : *B* → *C*. (Note a further crucial difference between **Ab** and **Grp**: in **Grp** it can happen that *f*(*A*) is not a normal subgroup of *B*, and that therefore the quotient group *B* / *f*(*A*) cannot be formed.) With these concrete descriptions of kernels and cokernels, it is quite easy to check that **Ab** is indeed an abelian category.

The product in **Ab** is given by the product of groups, formed by taking the cartesian product of the underlying sets and performing the group operation componentwise. Because **Ab** has kernels, one can then show that **Ab** is a complete category. The coproduct in **Ab** is given by the direct sum; since **Ab** has cokernels, it follows that **Ab** is also cocomplete.

We have a forgetful functor **Ab** → **Set** which assigns to each abelian group the underlying set, and to each group homomorphism the underlying function. This functor is faithful, and therefore **Ab** is a concrete category. The forgetful functor has a left adjoint (which associates to a given set the free abelian group with that set as basis) but does not have a right adjoint.

Taking direct limits in **Ab** is an exact functor. Since the group of integers **Z** serves as a generator, the category **Ab** is therefore a Grothendieck category; indeed it is the prototypical example of a Grothendieck category.

An object in **Ab** is injective if and only if it is a divisible group; it is projective if and only if it is a free abelian group. The category has a projective generator (**Z**) and an injective cogenerator (**Q**/**Z**).

Given two abelian groups *A* and *B*, their tensor product *A*⊗*B* is defined; it is again an abelian group. With this notion of product, **Ab** is a symmetric monoidal category.

**Ab** is not cartesian closed (and therefore also not a topos) since it lacks exponential objects.

## See also

- Category of modules
- Abelian sheaf - many facts about the category of abelian groups continue to hold for the category of sheaves of abelian groups

## References

- Pedicchio & Tholen (2004) p.200
- Mac Lane (1998) p.209

- Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics,**211**(Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 - Mac Lane, Saunders (1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics.**5**(2nd ed.). New York, NY: Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001. - Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004).
*Categorical foundations. Special topics in order, topology, algebra, and sheaf theory*. Encyclopedia of Mathematics and Its Applications.**97**. Cambridge: Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001.