Todd Trimble
monomorphisms in the category of groups

This page is to record a constructive proof of the following result.

Proposition

Every monomorphism in the category of groups is an equalizer.

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. Passing to the wreath product A GGA^G \rtimes G, we have two homomorphisms ϕ,ψ:GA GG\phi, \psi: G \rightrightarrows A^G \rtimes G defined by ϕ(g):=(d jπ(g),g)\phi(g) := (d_{j \pi}(g), g) and ψ(g):=(0,g)\psi(g) := (0, g). I 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; I chose the canonical one.)

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 the 𝔽 2\mathbb{F}_2-vector space freely 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 are the same, hence 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 epic implies ii is a epic mono. Epic monos ii being isomorphisms, ff is then forced to be regular epic as well.

Revised on June 28, 2017 at 10:20:07 by Todd Trimble