category of abelian groups

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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...
Ab has the
abelian group In mathematics, an abelian group, also called a commutative group, is a group (mathematics), group in which the result of applying the group Operation (mathematics), operation to two group elements does not depend on the order in which they are w ...
s as objects and
group homomorphism In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...
s as
morphism In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
s. This is the prototype of an abelian category: indeed, every
small Small may refer to: Science and technology * SMALL, an ALGOL-like programming language * Small (anatomy), the lumbar region of the back * Small (journal), ''Small'' (journal), a nano-science publication * HTML_element#Presentation, <small>, a ...
abelian category can be embedded in Ab.

# Properties

The zero object of Ab is the
trivial group In mathematics, a trivial group or zero group is a Group (mathematics), group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element ...
which consists only of its
neutral element In mathematics, an identity element, or neutral element, of a binary operation operating on a Set (mathematics), set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in alge ...
. The
monomorphism In the context of abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, fiel ...
s in Ab are the
injective In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
group homomorphisms, the epimorphisms are the
surjective In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
group homomorphisms, and the
isomorphism In mathematics, an isomorphism is a structure-preserving Map (mathematics), mapping between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are iso ...
s are the
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...
group homomorphisms. Ab is a
full subcategory In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
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 In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category (mathematics), category that is enriched category, enriched over the category of abelian groups, Ab. That is, an Ab-cate ...
, and because the
direct sum The direct sum is an Operation (mathematics), operation between Mathematical structure, structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct ...
of finitely many abelian groups yields a
biproduct In category theory and its applications to mathematics, a biproduct of a finite collection of Object (category theory), objects, in a category (mathematics), category with zero objects, is both a product (category theory), product and a coproduct. ...
, we indeed have an
additive category In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts. Definition A category C is preadditive if all its hom-sets are abelian groups and composition of morp ...
. 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
cokernel The cokernel of a linear mapping of vector spaces is the quotient space (linear algebra), quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual (category theory), d ...
s; the cokernel of ''f'' is the
quotient group A quotient group or factor group is a mathematical group (mathematics), group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factor ...
''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 inner automorphism, conjugation by members of the Group (mathematics), group of which it is a part. In o ...
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 Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Pr ...
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 Set (mathematics), sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notatio ...
of the underlying sets and performing the group operation componentwise. Because Ab has kernels, one can then show that Ab is a
complete category In mathematics, a complete category is a category (mathematics), category in which all small limit (category theory), limits exist. That is, a category ''C'' is complete if every diagram (category theory), diagram ''F'' : ''J'' → ''C'' (where ''J' ...
. The
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of Set (mathematics), sets and disjoint union (topology), of topological spaces, the free product of Group (mathematics), group ...
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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
. The forgetful functor has a
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
(which associates to a given set the
free abelian group In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation (mathematics), operation that is associative, commutative, and inverti ...
with that set as basis) but does not have a right adjoint. Taking
direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be Group (mathematics), groups, Ring (mathematics), rings, Vector spac ...
s 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
if and only if it is a divisible group; it is projective if and only if it is a
free abelian group In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation (mathematics), operation that is associative, commutative, and inverti ...
. 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 and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
''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 In mathematics, a topos (, ; plural topoi or , or toposes) is a category (mathematics), category that behaves like the category of Sheaf (mathematics), sheaves of Set (mathematics), sets on a topological space (or more generally: on a Site (math ...
since e.g. it has a zero object.

# See also

*
Category of modules In Abstract algebra, algebra, given a Ring (mathematics), ring ''R'', the category of left modules over ''R'' is the Category (mathematics), category whose Object (category theory), objects are all left Module (mathematics), modules over ''R'' and ...
* 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 Group theory