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I seem to remember reading somewhere an "abstract nonsense" proof of the well-known fact that the fundamental group of a connected topological group is abelian. By "abstract nonsense" I mean that the proof used little more than the fact that topological groups are the group objects in the category of topological spaces and the fact that $\pi_1$ is a homotopy functor. Does anybody remember how this works? References are fine.

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Google the "Eckman-Hilton argument". – Ryan Budney Dec 8 '10 at 1:43
@Ryan: I'm sure you're right. Would you perhaps consider leaving this as answer (possibly with a little more information and/or a link to a nice reference)? – Pete L. Clark Dec 8 '10 at 2:11
up vote 6 down vote accepted

There is a mostly nonsensical proof in Robert M. Switzer's Algebraic topology--homotopy and homology, at the end of the first chapter. (You can see it in googlebooks)

He proves that if $X$ is an $H$-cogroup, then $[X,Y]$ is a group, and that if $Y$ is an $H$-group, then $[X,Y]$ is also a group, so when $X$ and $Y$ are an $H$-cogroups and an $H$-group, respectively, then $[X,Y]$ is a group in two ways. And so on.

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This is more generally true in any category (replace H group or H cogroup with group and cogroup object--in the present case this is the pointed homotopy category of topological spaces). As Ryan Budney observes above, it is basically a consequence of the Eckmann-Hilton argument. – Akhil Mathew Dec 8 '10 at 2:25
@Akhil: it is the Eckmann Hilton argument. The intention of the answer is not to say anything original nor mnaximally general... just to point to a nonsensically enough proof. – Mariano Suárez-Alvarez Dec 8 '10 at 2:26

I just found this old question and happen to know a very nice "abstract nonsense" argument of a completely different flavor. Let $G$ be a topological group and consider the fundamental groupoid $\Pi(G)$ of $G$, which becomes a monoidal category under the group operation. The unit object is of of course the identity. But it is well known that the endomorphism monoid of the unit object, which in this case is the fundamental group at the identity, embeds into the Bernstein center (the monoid of endomorphisms of the identity functor on $\Pi(G)$), which is always commutative.

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