Category of abelian groups

Short description: Category whose objects are abelian groups and whose morphisms are group homomorphisms

In mathematics, the category $\mathbf{𝐀}\mathbf{𝐛}$ has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in $\mathbf{𝐀}\mathbf{𝐛}$.

Properties

The zero object of $\mathbf{𝐀}\mathbf{𝐛}$ is the trivial group $\{0\}$ which consists only of its neutral element.

The monomorphisms in $\mathbf{𝐀}\mathbf{𝐛}$ are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms.

$\mathbf{𝐀}\mathbf{𝐛}$ is a full subcategory of $\mathbf{𝐆}\mathbf{𝐫}\mathbf{𝐩}$, the category of all groups. The main difference between $\mathbf{𝐀}\mathbf{𝐛}$ and $\mathbf{𝐆}\mathbf{𝐫}\mathbf{𝐩}$ 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 $\mathbf{𝐀}\mathbf{𝐛}$ into a preadditive category, and because the direct sum of finitely many abelian groups yields a biproduct, we indeed have an additive category.

In $\mathbf{𝐀}\mathbf{𝐛}$, the notion of kernel in the category theory sense coincides with kernel in the algebraic sense, i.e. the categorical kernel of the morphism $f:A\to B$ is the subgroup $K$ of $A$ defined by $K=\{x\in A:f(x)=0\}$, together with the inclusion homomorphism $i:K\to A$. The same is true for cokernels; the cokernel of f is the quotient group $C=B/f(A)$ together with the natural projection $p:B\to C$. (Note a further crucial difference between $\mathbf{𝐀}\mathbf{𝐛}$ and $\mathbf{𝐆}\mathbf{𝐫}\mathbf{𝐩}$: in $\mathbf{𝐆}\mathbf{𝐫}\mathbf{𝐩}$ 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 $\mathbf{𝐀}\mathbf{𝐛}$ is indeed an abelian category.

The forgetful functor from $\mathbb{\mathbb{Z}}\text{-}\mathbf{𝐌}\mathbf{𝐨}\mathbf{𝐝}$ to $\mathbf{𝐀}\mathbf{𝐛}$ that sends a $\mathbb{\mathbb{Z}}$-module $(M,+,\cdot )$ to its underlying abelian group $(M,+)$ and the functor from $\mathbf{𝐀}\mathbf{𝐛}$ to $\mathbb{\mathbb{Z}}$ that sends an abelian group $(G,+)$ to the $\mathbb{\mathbb{Z}}$-module $(G,+,\cdot )$ obtained by setting $k\cdot g:=g^k$ define an isomorphism of categories.

The product in $\mathbf{𝐀}\mathbf{𝐛}$ is given by the product of groups, formed by taking the Cartesian product of the underlying sets and performing the group operation componentwise. Because $\mathbf{𝐀}\mathbf{𝐛}$ has kernels, one can then show that $\mathbf{𝐀}\mathbf{𝐛}$ is a complete category. The coproduct in $\mathbf{𝐀}\mathbf{𝐛}$ is given by the direct sum; since $\mathbf{𝐀}\mathbf{𝐛}$ has cokernels, it follows that $\mathbf{𝐀}\mathbf{𝐛}$ is also cocomplete.

We have a forgetful functor $\mathbf{𝐀}\mathbf{𝐛}\to \mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}$ which assigns to each abelian group the underlying set, and to each group homomorphism the underlying function. This functor is faithful, and therefore $\mathbf{𝐀}\mathbf{𝐛}$ 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 $\mathbf{𝐀}\mathbf{𝐛}$ is an exact functor. Since the group of integers $\mathbb{\mathbb{Z}}$ serves as a generator, the category $\mathbf{𝐀}\mathbf{𝐛}$ is therefore a Grothendieck category; indeed it is the prototypical example of a Grothendieck category.

An object in $\mathbf{𝐀}\mathbf{𝐛}$ 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 ($\mathbb{\mathbb{Z}}$) and an injective cogenerator ($\mathbb{\mathbb{Q}}/\mathbb{\mathbb{Z}}$).

Given two abelian groups $A$ and $B$, their tensor product $A\otimes B$ is defined; it is again an abelian group. With this notion of product, $\mathbf{𝐀}\mathbf{𝐛}$ is a closed symmetric monoidal category.

$\mathbf{𝐀}\mathbf{𝐛}$ is not a topos since e.g. it has a zero object.

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

• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4 
• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. 5 (2nd ed.). Springer. ISBN 0-387-98403-8. 
• 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 University Press. ISBN 0-521-83414-7.