- Category of abelian groups
In

mathematics , the category**Ab**has theabelian group s as objects andgroup homomorphism s asmorphism s. This is the prototype of anabelian category .The

monomorphism s in**Ab**are theinjective group homomorphisms, theepimorphism s are thesurjective group homomorphisms, and theisomorphism s are thebijective group homomorphisms.The

zero object of**Ab**is the trivial group {0} which consists only of itsneutral element .Note that

**Ab**is afull 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 apreadditive category , and because thedirect sum of finitely many abelian groups yields abiproduct , we indeed have anadditive category .In

**Ab**, the notion of kernel in the category theory sense coincides with kernel in the algebraic sense, i.e.: the kernel of the morphism "f" : "A" → "B" is the subgroup "K" of "A" defined by "K" = {"x" in "A" : "f"("x") = 0}, together with the inclusion homomorphism "i" : "K" → "A". The same is true forcokernel s: the cokernel of "f" is thequotient 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 anormal 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 anabelian category .The product in

**Ab**is given by the product of groups, formed by taking thecartesian product of the underlying sets and performing the group operation componentwise. Because**Ab**has kernels, one can then show that**Ab**is acomplete category . Thecoproduct in**Ab**is given by thedirect sum of groups; since**Ab**has cokernels, it follows that**Ab**is alsococomplete .Taking

direct limit s in**Ab**is anexact functor , which turns**Ab**into anAb5 category .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 aconcrete category . The forgetful functor has a left adjoint (which associates to a given set thefree abelian group with that set as basis) but does not have a right adjoint.An object in

**Ab**is injective if and only if it is divisible; it is projective if and only if it is a free abelian group. The category has a projective generator (**Z**) and aninjective 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 strict monoidal category.**Ab**is not cartesian closed (and therefore also not atopos ) since it lacksexponential object s.

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