How should I think about delooping?

Question

When talking about the Eilenberg-Maclane space $K(G,n)$, we usually restrict our attention to the situation where $G$ is abelian. In that case, we get $\Omega K(G,n)=K(G,n-1)$, so we can call $K(G,n)$ a delooping of $K(G,n-1)$.

Since $\pi_n$ is always abelian for $n>1$, it only makes sense to talk about $K(G,1)=BG$ for $G$ nonabelian anyways. So there definitely shouldn't be delooping of this space, because then it would have $\pi_2=G$, which is impossible. From the previous paragraph, it seems like we should therefore be able to say that the nonabelianity of $G$ (i.e., the nontriviality of the commutator $[G,G]$) is the obstruction to delooping $BG$. But this isn't very satisfying, because I can't quite see what's going on with the actual space.

All of which motivates my (slightly open-ended/up-to-interpretation) question:

How should I think about delooping? Is it nothing more than thing like "for the space $X$ that we care about, it just so happens that we've got $Y$ with $\Omega Y\simeq X$", or is there a definite way to measure obstructions? In the cases where a delooping exists, is there an explicit method for its construction?

Answer 1

One possible answer: Stasheff proved that a (connected) space $X$ is (homotopy equivalent to) a loopspace if and only if $X$ is an algebra over the $A_\infty$ operad (or rather I should say an $A_\infty$ operad).

See for instance this article.

Answer 2

I'm not sure whether you'll like this, but my natural response to "how should I think about delooping?" is to invoke (higher) category theory. You may know that a homotopy 1-type, i.e. a space (probably a CW complex) with $\pi_n=0$ for n>1, is uniquely specified up to (weak) homotopy equivalence by its fundamental groupoid. In fact, one has an equivalence of (2- or homotopy-) categories, so we can identify homotopy 1-types with groupoids. Under this identification, the space BG which deloops a discrete group G is identified with the groupoid with one object and G as the automorphism group of that object. So one-step delooping of a discrete group really is just the simple process of considering a group as a one-object groupoid (although in the homotopy theory world it requires a fairly elaborate construction).

At higher levels, the "homotopy hypothesis" in higher category theory (which is a theorem for some definitions of higher category) says that homotopy n-types can be identified with n-groupoids, and arbitrary homotopy types with ∞-groupoids. Moreover, the identification of groups with one-object groupoids is believed to continue to higher categories as well: 2-groups (i.e. groupoids equipped with an extra group structure up to coherent isomorphism) can be identified with one-object 2-groupoids, and similarly an n-group can be identified with a one-object n-groupoid. Thus, deloopability of a space requires that it be equipped with a suitable group structure (up to homotopy, i.e. up to equivalence), and in that case its delooping corresponds on the categorical side to regarding an n-group as a one-object n-groupoid (where possibly n=∞).

Finally, as to the obstruction to delooping BG when G is not abelian, only when G is abelian is BG itself a (2-)group. The reason is the same one that other people have mentioned—Eckmann-Hilton—but I prefer to think about it in these terms.