A few weeks ago on MathOverflow Greg Muller asked, “why do groups and abelian groups feel so different?” The answers were very interesting and came from several different perspectives, but I still don’t feel as if the question was resolved. So I’ll try to synthesize and summarize some of the answers and hopefully something will be clearer in the end.
Groups naturally arise as automorphism groups of objects in categories. From this perspective abelian groups should just correspond to objects with particularly simple symmetries. But abelian groups have their own special type of symmetry: the pointwise sum of two homomorphisms is also a homomorphism, so is enriched over itself, something that doesn’t hold at all for . What this implies is that the automorphisms of an abelian group have two composition structures: one coming from pointwise addition and another coming from composition. This is the “best” definition of a ring, at least in my opinion.
This observation also immediately tells us why is a special ring: it is the initial object in , since every abelian group has a -action given by . From here one is readily led to the “best” definition of a module, exactly analogous to how one might go from sets to permutations to groups to group actions. Now we can think of abelian groups as -modules and the action as a distinguished set of automorphisms of any abelian group. For some abelian groups these are the only automorphisms.
Non-abelian groups, on the other hand, have a totally different set of distinguished automorphisms: the inner automorphisms . Indeed, one way to define a non-abelian group is as a group with nontrivial inner automorphisms, and for some non-abelian groups these are the only automorphisms. In other words, while the structure of abelian groups is controlled in a strong way by the structure of (this is basically the content of the structure theorem), there is no such controlling object for general groups.
Perhaps precisely because their structure is so well-controlled, abelian groups lead to a lot of useful constructions in mathematics, such as abelian categories (the basic tool of homological algebra). Even to study groups one often passes to the category of modules over the group algebra!
What’s not clear to me is the following: should I think of abelianness as a powerful tool or as the easier version of a more difficult but more powerful theory of “non-abelian categories”?
A 2-categorical perspective
One particular complaint Greg had was that abelian groups rarely arise, in practice, as automorphism groups of objects. There is a way, however, one can get abelian groups “naturally” acting on things. The Eckmann-Hilton argument shows that two monoid structures on a set which are homomorphisms of each other must in fact be the same commutative monoid structure, and has the following consequences:
- The higher homotopy groups are abelian.
- The -morphisms in a -category with one object and one morphism form a commutative monoid. If the -morphisms are invertible, they form an abelian group.
The latter is due to the fact that -categories have two types of composition. This seems to have important philosophical implications; for example, awhile back at the n-Category cafe Bruce Bartlett asked a similar question about the difference between commutative and noncommutative algebras from an -categorical point of view. It seems as if to understand commutativity one should really move up the -categorical ladder; as Scott Carnahan puts it, if groups act on objects, then 2-groups act on categories, and the automorphisms of the identity functor automatically form an abelian group.
The big question, of course, is whether the abelian groups that naturally arise in mathematics can actually be fruitfully thought of in this way. Anyone have any thoughts?