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.
The monomorphisms in 'Ab' are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms.
The zero object of 'Ab' is the trivial group {0} which consists only of its neutral element.
Note that '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 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 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 of groups; since 'Ab' has cokernels, it follows that 'Ab' is also cocomplete.
Taking direct limits in 'Ab' is an exact functor, which turns 'Ab' into an Ab5 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 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.
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 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 strict monoidal category.
'Ab' is not cartesian closed (and therefore also not a topos) since it lacks exponential objects.