nLab
Grp

Contents

Definition

GrpGrp is the category with groups as objects and group homomorphisms as morphisms.

More abstractly, we can think of GrpGrp as the full subcategory of CatCat with groups as objects.

If EE is any category with finite products, there is a category Grp(E)Grp(E) of group objects in EE. This category if equivalent to the category Prod(T Grp,E)Prod(T_{Grp}, E) of product-preserving functors from the Lawvere theory for groups to EE.

Remarks

Since groups may be identified with one-object groupoids, it is sometimes useful to regard GrpGrp as a 22-category, namely as the full sub-22-category of Grpd on one-object groupoids. In this case the 22-morphisms between homomorphisms of groups come from “intertwiners”: inner automorphisms of the target group.

On the other hand, if we regard GrpGrp as a full sub-22-category of Grpd *Grpd_*, the 22-category of pointed groups, then this is locally discrete and equivalent to the ordinary 11-category GrpGrp. This is because the only pointed intertwiner between two homomorphisms is the identity.

Precisely analogous statements hold for the category Alg of algebras.

Properties

(In this section, all statements about GrpGrp are valid more generally for Grp(E)Grp(E) where EE is a topos with a natural numbers object.)

The category GrpGrp is one of the prototypical examples of a semiabelian category, and so enjoys some nice properties. For example, it is regular and even exact, and protomodular so that one can expect a certain battery of diagram chasing lemmas to hold in it.

The category of groups is also balanced. This follows from a somewhat remarkable theorem:

Theorem

Every monomorphism in GrpGrp is an equalizer.

The proofs most commonly seen in the literature are elementary but nonconstructive; a typical example may be found here at regular monomorphism. Here we give a constructive proof.

Proof

Let i:HGi: H \to G be monic, and let π:GG/H\pi: G \to G/H be the canonical surjective function ggHg \mapsto g H. Let AA be the free abelian group on G/HG/H with j:G/HAj: G/H \to A the canonical injection, and let A GA^G denote the set of functions f:GAf: G \to A, with the pointwise abelian group structure inherited from AA. This carries a GG-module structure defined by

(gf)(g)=f(gg 1).(g \cdot f)(g') = f(g' g^{-1}).

For any fA Gf \in A^G, the function d f:GA Gd_f: G \to A^G defined by d f(g)=gffd_f(g) = g f - f defines a derivation, i.e., a map satisfying the equation d f(gg)=gd f(g)+d f(g)d_f(g g') = g d_f(g') + d_f(g). Consider now the wreath product, i.e., the semidirect product A GGA^G \rtimes G, where the multiplication is defined by (f,g)(f,g)=(f+gf,gg)(f, g) \cdot (f', g') = (f + g f', g g'). By the derivation equation, we have a homomorphism ϕ:GA GG\phi: G \to A^G \rtimes G defined by ϕ(g)(d jπ(g),g)\phi(g) \coloneqq (d_{j \pi}(g), g), and there is a second homomorphism ψ:GA GG\psi: G \to A^G \rtimes G defined by ψ(g)(0,g)\psi(g) \coloneqq (0, g). We claim that i:HGi: H \to G is the equalizer of the pair ϕ,ψ\phi, \psi. For,

d jπ(g)=0 iff ( g:G)gjπ(g)=jπ(g) iff ( g:G)jπ(gg 1)=jπ(g) iff ( g:G)j(gg 1H)=j(gH) iff ( g:G)gg 1H=gH iff g 1H=H iff gH.\array{ d_{j\pi}(g) = 0 & \text{iff} & (\forall_{g': G})\; g\cdot j\pi(g') = j\pi(g') \\ & \text{iff} & (\forall_{g': G})\; j\pi(g' g^{-1}) = j\pi(g') \\ & \text{iff} & (\forall_{g': G})\; j(g' g^{-1}H) = j(g' H) \\ & \text{iff} & (\forall_{g': G})\; g' g^{-1} H = g' H \\ & \text{iff} & g^{-1} H = H \\ & \text{iff} & g \in H. }

(All we needed was some injection j:G/HAj: G/H \to A into an abelian group; embedding into the free abelian group is a pretty canonical choice.)

Remark

This proof can be adapted to show that monomorphisms in the category of finite groups (group objects in FinSetFinSet) are also equalizers. All that needs to be modified is the choice of AA, which we could take to be a free 𝔽 2\mathbb{F}_2-vector space generated by G/HG/H.

Corollary

The category of groups is balanced: every epic mono is an isomorphism.

Proof

This follows because an epic equalizer is an equalizer of two maps that (by epi-ness) must be the same, hence the equalizer is an isomorphism.

Corollary

Every epimorphism in the category of groups is a coequalizer.

Proof

Since every morphism f:GHf: G \to H factors as a regular epi p:GG/ker(f)p: G \to G/\ker(f) followed by a mono ii, having ff epi implies ii is a epic mono. Epic monos ii being isomorphisms, ff is then forced to be regular epic as well.

Remark

Despite the fact that every morphism in GrpGrp factors as an epi followed by a regular mono, it is not true that Grp opGrp^{op} is regular. Indeed, (regular) monos are in GrpGrp not stable under pushouts. This follows essentially from the plenitude of simple objects in GrpGrp: if HH is not simple but embeds in a simple group GG, then there is a nontrivial quotient HH/NH \to H/N, and in the pushout diagram

H G H/N P\array{ H & \hookrightarrow & G \\ \downarrow & & \downarrow \\ H/N & \to & P }

the object PP will be a proper quotient of GG and therefore P1P \cong 1, so that the pushout of the mono HGH \to G which is H/N1H/N \to 1 fails to be mono.

category: category

Revised on June 28, 2017 07:29:08 by Todd Trimble (24.146.226.222)