Difference between revisions of "Category of groups"

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In [[mathematics]], the [[category theory|category]] '''Grp''' has the [[class (set theory)|class]] of all [[Group (mathematics)|groups]] for objects and [[group homomorphism]]s for [[morphism]]s. As such, it is a [[concrete category]]. The study of this category is known as [[group theory]].


There are two forgetful functors from '''Grp''':
In [[mathematics]], the [[Category theory|category]] '''Grp''' has the [[Class (set theory)|class]] of all [[Group (mathematics)|groups]] for objects and [[group homomorphism]]s for [[morphism]]s. As such, it is a [[concrete category]]. The study of this category is known as [[group theory]].
 
==Relation to other categories==
There are two [[forgetful functor]]s from '''Grp''':


M:'''Grp''' → '''Mon'''
M:'''Grp''' → '''Mon'''
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The forgetful functor U:'''Grp''' → '''Set''' have a left adjoint given by the composite KF:'''Set'''→'''Mon'''→'''Grp''' where F is the free functor.
The forgetful functor U:'''Grp''' → '''Set''' have a left adjoint given by the composite KF:'''Set'''→'''Mon'''→'''Grp''' where F is the free functor.


 
==Categorical properties==
 
 
 
The [[monomorphism]]s in '''Grp''' are precisely the [[injective]] homomorphisms, the [[epimorphism]]s are precisely the [[surjective]] homomorphisms, and the [[isomorphism]]s are precisely the [[bijective]] homomorphisms.
The [[monomorphism]]s in '''Grp''' are precisely the [[injective]] homomorphisms, the [[epimorphism]]s are precisely the [[surjective]] homomorphisms, and the [[isomorphism]]s are precisely the [[bijective]] homomorphisms.


The category '''Grp''' is both [[complete category|complete and co-complete]]. The [[product (category theory)|category-theoretical product]] in '''Grp''' is just the [[direct product of groups]] while the [[coproduct|category-theoretical coproduct]] in '''Grp''' is the [[free product]] of groups. The [[zero object]]s in '''Grp''' are the [[trivial group]]s (consisting of just an identity element).
The category '''Grp''' is both [[complete category|complete and co-complete]]. The [[product (category theory)|category-theoretical product]] in '''Grp''' is just the [[direct product of groups]] while the [[coproduct|category-theoretical coproduct]] in '''Grp''' is the [[free product]] of groups. The [[zero object]]s in '''Grp''' are the [[trivial group]]s (consisting of just an identity element).


Every morphism ''f'' : ''G'' → ''H'' in '''Grp''' has a [[kernel (category theory)|category-theoretic kernel]] (given by the ordinary [[kernel (algebra)|kernel of algebra]] ker f = {''x'' in ''G'' | ''f''(''x'') = ''e''}), and also a [[cokernel (category theory)|category-theoretic cokernel]] (given by the [[factor group]] of ''H'' by the [[Normal closure (group theory)|normal closure]] of ''f''(''H'') in ''H''). Unlike in abelian categories, it is not true that every monomorphism in '''Grp''' is the kernel of its cokernel.
==As a non-abelian category==
The [[category of abelian groups]], '''Ab''', is a [[full subcategory]] of '''Grp'''. '''Ab''' is an [[abelian category]], but '''Grp''' is not. Indeed, '''Grp''' isn't even an [[additive category]], because there is no natural way to define the "sum" of two group homomorphisms. (The set of morphisms from the [[symmetric group]] ''S''<sub>3</sub> of order three to itself, <math>E=\operatorname{Hom}(S_3,S_3)</math>, has ten elements: an element ''z'' whose product on either side with every element of ''E'' is ''z'' (the homomorphism sending every element to the identity), three elements such that their product on one fixed side is always itself (the projections onto the three subgroups of order two), and six automorphisms.  If '''Grp''' were an additive category, then this set ''E'' of ten elements would be a [[ring (mathematics)|ring]].  In any ring, the zero element is singled out by the property that 0''x''=''x''0=0 for all ''x'' in the ring, and so ''z'' would have to be the zero of ''E''.  However, there are no two nonzero elements of ''E'' whose product is ''z'', so this finite ring would have no [[zero divisor]]s.  A [[finite ring]] with no zero divisors is a [[field (mathematics)|field]], but there is no field with ten elements because every [[finite field]] has for its order, the power of a prime.)
The [[category of abelian groups]], '''Ab''', is a [[full subcategory]] of '''Grp'''. '''Ab''' is an [[abelian category]], but '''Grp''' is not. Indeed, '''Grp''' isn't even an [[additive category]], because there is no natural way to define the "sum" of two group homomorphisms. (The set of morphisms from the [[symmetric group]] ''S''<sub>3</sub> of order three to itself, <math>E=\operatorname{Hom}(S_3,S_3)</math>, has ten elements: an element ''z'' whose product on either side with every element of ''E'' is ''z'' (the homomorphism sending every element to the identity), three elements such that their product on one fixed side is always itself (the projections onto the three subgroups of order two), and six automorphisms.  If '''Grp''' were an additive category, then this set ''E'' of ten elements would be a [[ring (mathematics)|ring]].  In any ring, the zero element is singled out by the property that 0''x''=''x''0=0 for all ''x'' in the ring, and so ''z'' would have to be the zero of ''E''.  However, there are no two nonzero elements of ''E'' whose product is ''z'', so this finite ring would have no [[zero divisor]]s.  A [[finite ring]] with no zero divisors is a [[field (mathematics)|field]], but there is no field with ten elements because every [[finite field]] has for its order, the power of a prime.)


Every morphism ''f'' : ''G'' → ''H'' in '''Grp''' has a [[kernel (category theory)|category-theoretic kernel]] (given by the ordinary [[kernel (algebra)|kernel of algebra]] ker f = {''x'' in ''G'' | ''f''(''x'') = ''e''}), and also a [[cokernel (category theory)|category-theoretic cokernel]] (given by the [[factor group]] of ''H'' by the [[Normal closure (group theory)|normal closure]] of ''f''(''H'') in ''H''). Unlike in abelian categories, it is not true that every monomorphism in '''Grp''' is the kernel of its cokernel.
==Exact sequences==
 
The notion of [[exact sequence]] is meaningful in '''Grp''', and some results from the theory of abelian categories, such as the [[nine lemma]], the [[five lemma]], and their consequences hold true in '''Grp'''. The [[snake lemma]] however is not true in '''Grp'''.
The notion of [[exact sequence]] is meaningful in '''Grp''', and some results from the theory of abelian categories, such as the [[nine lemma]], the [[five lemma]], and their consequences hold true in '''Grp'''. The [[snake lemma]] however is not true in '''Grp'''.


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[[Category:Category-theoretic categories|Groups]]
[[Category:Category-theoretic categories|Groups]]
[[Category:Group theory]]
[[Category:Group theory]]
[[es:Categoría de grupos]]
[[nl:Categorie van groepen]]

Latest revision as of 20:15, 22 November 2014

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} Template:Group theory sidebar

In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.

Relation to other categories

There are two forgetful functors from Grp:

M:GrpMon

U:GrpSet

Where M has two adjoints:

One right; I:MonGrp

One left; K:MonGrp

Here I:MonGrp is the functor sending every monoid to the submonoid of invertible elements and K:MonGrp the functor sending every monoid to the Grothendieck group of that monoid.

The forgetful functor U:GrpSet have a left adjoint given by the composite KF:SetMonGrp where F is the free functor.

Categorical properties

The monomorphisms in Grp are precisely the injective homomorphisms, the epimorphisms are precisely the surjective homomorphisms, and the isomorphisms are precisely the bijective homomorphisms.

The category Grp is both complete and co-complete. The category-theoretical product in Grp is just the direct product of groups while the category-theoretical coproduct in Grp is the free product of groups. The zero objects in Grp are the trivial groups (consisting of just an identity element).

Every morphism f : GH in Grp has a category-theoretic kernel (given by the ordinary kernel of algebra ker f = {x in G | f(x) = e}), and also a category-theoretic cokernel (given by the factor group of H by the normal closure of f(H) in H). Unlike in abelian categories, it is not true that every monomorphism in Grp is the kernel of its cokernel.

As a non-abelian category

The category of abelian groups, Ab, is a full subcategory of Grp. Ab is an abelian category, but Grp is not. Indeed, Grp isn't even an additive category, because there is no natural way to define the "sum" of two group homomorphisms. (The set of morphisms from the symmetric group S3 of order three to itself, , has ten elements: an element z whose product on either side with every element of E is z (the homomorphism sending every element to the identity), three elements such that their product on one fixed side is always itself (the projections onto the three subgroups of order two), and six automorphisms. If Grp were an additive category, then this set E of ten elements would be a ring. In any ring, the zero element is singled out by the property that 0x=x0=0 for all x in the ring, and so z would have to be the zero of E. However, there are no two nonzero elements of E whose product is z, so this finite ring would have no zero divisors. A finite ring with no zero divisors is a field, but there is no field with ten elements because every finite field has for its order, the power of a prime.)

Exact sequences

The notion of exact sequence is meaningful in Grp, and some results from the theory of abelian categories, such as the nine lemma, the five lemma, and their consequences hold true in Grp. The snake lemma however is not true in Grp.

References

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