Grp

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

More abstractly, we can think of $Grp$ as the full subcategory of $Cat$ with groups as objects.

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

Since groups may be identified with one-object groupoids, it is sometimes useful to regard $Grp$ 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.

On the other hand, if we regard $Grp$ as a full sub-$2$-category of $Grpd_*$, the $2$-category of *pointed* groups, 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.

(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:

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.

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

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

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

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

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

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 $G/H$.

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

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.

Every epimorphism in the category of groups is a coequalizer.

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.

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.

category: category

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