Cohomology and Extensions
Abelianisation is the process of freely making an algebraic structure ‘abelian’. There are several notions of abelianizations for various algebraic structures.
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.
The abelianization is an abelian group. Indeed, it is the universal abelian group induced by , in the following sense:
Hence abelianization is the free construction of an abelian group from a group.
Given a pointed connected topological space , its first singular homology group with coefficients in the integers is the abelianisation of its fundamental group:
This is a natural isomorphism filling the following diagram of functors:
(where forgets the point).
This example can also be done starting with an arbitrary pointed topological space and letting 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 is the abelianization of the free group on .
In other words, if is the free group-functor and is the free abelian group functor, then
commutes up to a canonical isomorphism. This is because we have a corresponding commutative diagram of forgetful functors
and is left adjoint to .
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.