In mathematics, the category <math>\mathbf{Ab}</math> 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 <math>\mathbf{Ab}</math>.
Properties
The zero object of <math>\mathbf{Ab}</math> is the trivial group <math>\{0\}</math> which consists only of its neutral element.
The monomorphisms in <math>\mathbf{Ab}</math> are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms.
<math>\mathbf{Ab}</math> is a full subcategory of <math>\mathbf{Grp}</math>, the category of all groups. The main difference between <math>\mathbf{Ab}</math> and <math>\mathbf{Grp}</math> is that the sum of two homomorphisms <math>f</math> and <math>g</math> between abelian groups is again a group homomorphism:
:<math>(f+g)(x+y)=f(x+y)+g(x+y)=f(x)+f(y)+g(x)+g(y)</math>
:<math>=f(x)+g(x)+f(y)+g(y)=(f+g)(x)+(f+g)(y)</math>
The third equality requires the group to be abelian. This addition of morphism turns <math>\mathbf{Ab}</math> into a preadditive category, and because the direct sum of finitely many abelian groups yields a biproduct, we indeed have an additive category.
In <math>\mathbf{Ab}</math>, the notion of kernel in the category theory sense coincides with kernel in the algebraic sense, i.e. the categorical kernel of the morphism <math>f:A\to B</math> is the subgroup <math>K</math> of <math>A</math> defined by <math>K=\{x\in A:f(x)=0\}</math>, together with the inclusion homomorphism <math>i:K\to A</math>. The same is true for cokernels; the cokernel of f is the quotient group <math>C=B/f(A)</math> together with the natural projection <math>p:B\to C</math>. (Note a further crucial difference between <math>\mathbf{Ab}</math> and <math>\mathbf{Grp}</math>: in <math>\mathbf{Grp}</math> it can happen that <math>f(A)</math> is not a normal subgroup of <math>B</math>, and that therefore the quotient group <math>B/f(A)</math> cannot be formed.) With these concrete descriptions of kernels and cokernels, it is quite easy to check that <math>\mathbf{Ab}</math> is indeed an abelian category.
The forgetful functor from <math>\mathbb{Z}\text{-}\mathbf{Mod}</math> to <math>\mathbf{Ab}</math> that sends a <math>\mathbb{Z}</math>-module <math>(M,+,\cdot)</math> to its underlying abelian group <math>(M,+)</math> and the functor from <math>\mathbf{Ab}</math> to <math>\mathbb{Z}</math> that sends an abelian group <math>(G,+)</math> to the <math>\mathbb{Z}</math>-module <math>(G,+,\cdot)</math> obtained by setting <math>k \cdot g := g^{k}</math> define an isomorphism of categories.
The product in <math>\mathbf{Ab}</math> is given by the product of groups, formed by taking the Cartesian product of the underlying sets and performing the group operation componentwise. Because <math>\mathbf{Ab}</math> has kernels, one can then show that <math>\mathbf{Ab}</math> is a complete category. The coproduct in <math>\mathbf{Ab}</math> is given by the direct sum; since <math>\mathbf{Ab}</math> has cokernels, it follows that <math>\mathbf{Ab}</math> is also cocomplete.
We have a forgetful functor <math>\mathbf{Ab}\to\mathbf{Set}</math> which assigns to each abelian group the underlying set, and to each group homomorphism the underlying function. This functor is faithful, and therefore <math>\mathbf{Ab}</math> 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 <math>\mathbf{Ab}</math> is an exact functor. Since the group of integers <math>\mathbb{Z}</math> serves as a generator, the category <math>\mathbf{Ab}</math> is therefore a Grothendieck category; indeed it is the prototypical example of a Grothendieck category.
An object in <math>\mathbf{Ab}</math> 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 (<math>\mathbb{Z}</math>) and an injective cogenerator (<math>\mathbb{Q}/\mathbb{Z}</math>).
Given two abelian groups <math>A</math> and <math>B</math>, their tensor product <math>A\otimes B</math> is defined; it is again an abelian group. With this notion of product, <math>\mathbf{Ab}</math> is a closed symmetric monoidal category.
<math>\mathbf{Ab}</math> 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