Category of abelian groups

In mathematics, the category ${\displaystyle \mathbf {Ab} }$ 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 ${\displaystyle \mathbf {Ab} }$. ^[2]

Properties

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

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

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

${\displaystyle (f+g)(x+y)=f(x+y)+g(x+y)=f(x)+f(y)+g(x)+g(y)}$

${\displaystyle =f(x)+g(x)+f(y)+g(y)=(f+g)(x)+(f+g)(y)}$

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

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

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

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

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

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

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

${\displaystyle \mathbf {Ab} }$ 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

1. Pedicchio & Tholen 2004 , p. 200
2. Mac Lane 1998 , p. 209

• Lang, Serge (2002), Algebra , Graduate Texts in Mathematics, vol. 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. Vol. 5 (2nd ed.). Springer. 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. Vol. 97. Cambridge University Press. ISBN   0-521-83414-7. Zbl   1034.18001.