nLab abelianization

Idea

Abelianisation is the process of freely making an algebraic structure ‘abelian’. There are several notions of abelianizations for various algebraic structures, notably there is the abelianization of non-abelian groups to abelian groups.

There is also Verdier's abelianization functor from a triangulated category to an abelian category with some universal property; this is treated in a separate entry.

For groups

Definition

Definition

For $G$ a group, its abelianization $G^{ab} \in$ Grp is the quotient of $G$ by its commutator subgroup:

$$G^{ab} \coloneqq G/[G,G] \,,$$

A group whose abelianization is the trivial group is called a perfect group.

The abelianization is an abelian group. Indeed, it is the universal abelian group induced by $G$, in the following sense:

Proposition

Abelianization extends to a functor $(-)^{ab} \colon$ Grp $\to$ Ab and this functor is left adjoint to the forgetful functor $U \colon Ab \to Grp$ from abelian groups to group.

Hence abelianization is the free construction of an abelian group from a group.

Examples

Homotopy groups

Example

Given a pointed connected topological space $(X,a)$, its first singular homology group with coefficients in the integers is the abelianisation of its fundamental group:

$$H_1(X,\mathbb{Z}) \cong \pi_1(X,a)^{ab} .$$

This is a natural isomorphism filling the following diagram of functors:

$$\array { Top_{\geq 1}^{*/} & \overset{\pi_1}\longrightarrow & Grp \\ \llap{U}\downarrow & & \downarrow\rlap{ab} \\ Top & \underset{H_1({-},\mathbb{Z})}\longrightarrow & Ab Grp }$$

(where $U$ forgets the point).

This example can also be done starting with an arbitrary pointed topological space and letting $U$ take the connected component of the point. Of course, this example really lives in ∞ Grpd and so we could start with a (pointed, maybe connected) simplicial set, Kan complex, etc.

For more discussion of this see at singular homology the section Relation to homotopy groups.

Free abelian groups

A free abelian group on a set $S$ is the abelianization of the free group on $S$.

In other words, if $F \colon Set \to Grp$ is the free group-functor and $F_{Ab} \colon Set \to Ab$ is the free abelian group functor, then

$$\array{ Set &&\stackrel{F_{ab}}{\to}&& Ab \\ & {}_{\mathllap{F_{grp}}}\searrow && \nearrow_{\mathrlap{(-)^{ab}}} \\ && Grp }$$

commutes up to a canonical isomorphism. This is because we have a corresponding commutative diagram of forgetful functors

$$\array{ Set &&\stackrel{U_{ab}}{\leftarrow}&& Ab \\ & {}_{\mathllap{U_{grp}}}\nwarrow && \swarrow_{\mathrlap{U}} \\ && Grp }$$

and $(-)^{ab} \circ F_{grp}$ is left adjoint to $U_{grp} \circ U$.

For monoids etc

Abelianisation of monoids works pretty much like abelianisation of groups.

We can also do abelianisation of monoid objects in many monoidal categories (or closed categories or more generally multicategories). For example, we can form abelianisations of rings, which are monoid objects in Ab.

We can even form abelianisations of semigroups or magmas.

For Lie algebras

Lie algebras are not monoid objects in any category, but one still considers abelian Lie algebras, which may be identified with their underlying vector spaces. These are so called because they correspond to abelian Lie groups. Lie algebras also can be abelianised.