category of abelian groups
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the category Ab has the abelian groups as objects and
group homomorphism In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) whe ...
s as morphisms. This is the prototype of an abelian category: indeed, every
small Small means of insignificant size Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to three geometrical measures: length, area, or ...
abelian category can be embedded in Ab.


Properties

The zero object of Ab is the trivial group which consists only of its neutral element. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
s are the bijective group homomorphisms. Ab is a
full subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...
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 categorical kernel of the morphism ''f'' : ''A'' → ''B'' is the subgroup ''K'' of ''A'' defined by ''K'' = , 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 In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
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 forgetful functor from \mathbbMod to Ab that sends a \mathbb-module (M,+,\cdot) to its underlying abelian group (M,+) and the functor from Ab to \mathbb that sends an abelian group (G,+) to the \mathbb-module (G,+,\cdot) obtained by setting k \cdot g := g^ define an isomorphism of categories. The product in Ab is given by the product of groups, formed by taking the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
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; since Ab has cokernels, it follows that Ab is also cocomplete. 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. Taking direct limits in Ab is an exact functor. Since the group of integers Z serves as a generator, the category Ab is therefore a Grothendieck category; indeed it is the prototypical example of a Grothendieck category. An object in 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 (Z) and an injective cogenerator (Q/Z). Given two abelian groups ''A'' and ''B'', their
tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
''A''⊗''B'' is defined; it is again an abelian group. With this notion of product, Ab is a closed symmetric monoidal category. 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

* * * {{DEFAULTSORT:Category Of Abelian Groups Abelian groups Abelian group theory