nLab Grp

Definition

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

Similarly there is the full subcategory $FinGrp \hookrightarrow Grp$ of finite groups.

More generally, if $\mathcal{C}$ is any category with finite products, there is a category $Grp(\mathcal{C})$ of group objects in $\mathcal{C}$. This category if equivalent to the category $Prod(T_{Grp}, \mathcal{C})$ of product-preserving functors from the Lawvere theory for groups to $\mathcal{C}$. For instance for $\mathcal{C} =$ SmthMfd this yields the category of Lie groups; while for $\mathcal{C} =$ Set it reduces again to the default case.

Remarks

If one associates to a group $G$ its delooping one-object groupoid $B G$, it is sometimes of interest to regard the collection of groups instead as a 2-category, namely as the full sub-$2$-category of Grpd on one-object groupoids. In this case the $2$-morphisms between homomorphisms of groups come from “intertwiners”: inner automorphisms of the target group – hence this 2-category is not equivalent to the 1-category of groups.

On the other hand, if we regard $Grp$ as a full sub-$2$-category of $Grpd_*$, the $2$-category of pointed groupoids, then this is locally discrete and equivalent to the ordinary 1-category $Grp$. 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 $Grp$ are valid more generally for $Grp(E)$ where $E$ is a topos with a natural numbers object.)

The category $Grp$ 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 $Grp$ 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 $i: H \to G$ be monic, and let $\pi: G \to G/H$ be the canonical surjective function $g \mapsto g H$. Let $A$ be the free abelian group on $G/H$ with $j: G/H \to A$ the canonical injection, and let $A^G$ denote the set of functions $f: G \to A$, with the pointwise abelian group structure inherited from $A$. This carries a $G$-module structure defined by

$$(g \cdot f)(g') = f(g' g).$$

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

$$\array{ d_{j\pi}(g) = 0 & \text{iff} & (\forall_{g': G})\; g\cdot j\pi(g') = j\pi(g') \\ & \text{iff} & (\forall_{g': G})\; j\pi(g' g) = j\pi(g') \\ & \text{iff} & (\forall_{g': G})\; j(g' g H) = j(g' H) \\ & \text{iff} & (\forall_{g': G})\; g' g H = g' H \\ & \text{iff} & g H = H \\ & \text{iff} & g \in H. }$$

(All we needed was some injection $j: G/H \to A$ 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 $FinSet$) are also equalizers. All that needs to be modified is the choice of $A$, which we could take to be a free $\mathbb{F}_2$-vector space generated by $G/H$.

Remark

The proof requires that the unit $X \to U F X$ of the free abelian group monad (or the free $\mathbb{F}_2$ vector space monad) is monic, constructively. This is not entirely obvious, and is not a well-known fact, but details can be found in this MO thread.

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 $f: G \to H$ factors as a regular epi $p: G \to G/\ker(f)$ followed by a mono $i$, having $f$ epi implies $i$ is a epic mono. Epic monos $i$ being isomorphisms, $f$ is then forced to be regular epic as well.

Remark

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

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

the object $P$ will be a proper quotient of $G$ and therefore $P \cong 1$, so that the pushout of the mono $H \to G$ which is $H/N \to 1$ fails to be mono.

Related entries

• Ab, TopGrp

References

An axiomatization of the category of groups is given in

• Pierre Leroux, Une Caracterisation de la Categorie des Groupes, Canadian Mathematical Bulletin, Volume 15, Issue 3, September 1972, pp. 375 - 380 (DOI:10.4153/CMB-1972-069-6)

category: category