Eckmann-Hilton and higher homotopy groups
The essential point is the following (I'll only talk about $\pi_2$ but everything I say here generalizes). Let $H : I^2 \to X$ be a continuous map, where $I = [0, 1]$ is the interval (so that $I^2$ is a square). Then $H$ may be regarded as a homotopy between two paths $I \to X$ in two ways:
- as a homotopy between the paths $H(0, -)$ and $H(1, -)$, or
- as a homotopy between the paths $H(-, 0)$ and $H(-, 1)$.
This induces two notions of composition called vertical and horizontal composition on such maps, one of which stacks the corresponding squares vertically (when the corresponding paths agree) and one of which stacks the squares horizontally. Both of these operations are associative up to homotopy, and the condition that each of them is a homomorphism for the other (up to homotopy) is precisely the statement that you can interpret the mushing of four squares together into a single big square in two different ways (stacking horizontally then vertically, or stacking vertically then horizontally) and these are the same thing. (This can be thought of as a $2$-dimensional statement in the vein of of the associativity of composition of functions, which is a $1$-dimensional statement about stacking lines.)
Now add the additional condition that all four sides of the square are mapped to the basepoint $x_0$. Then there is an identity square $i$ (for both vertical and horizontal composition) which sends everything to $x_0$, and running the Eckmann-Hilton argument here amounts to pushing squares around. In ASCII art form, it looks like this:
----- --------- --------- ----- | f | | i | f | --------- | g | i | | g | ----- = --------- = | g | f | = --------- = ----- | g | | g | i | --------- | i | f | | f | ----- --------- --------- -----
The above is really an argument in the $2$-category $\Pi_2(X)$. See my blog post Centers, 2-categories, and the Eckmann-Hilton argument for an elaboration on this. In the notation of that blog post the Eckmann-Hilton argument looks like this:
Recall that the Eckmann-Hilton argument says that if you have two different unital monoid structures on a set such that both are homomorphisms for the other and both have the same unit, then they're equal and commutative. We want to apply that to see that higher homotopy groups are commutative.
The two different monoid structures we want to consider are 'vertical' and 'horizontal' concatenation of maps from $I^n$: if I have two maps $f,g:I^n\to X$ for some topological space $X$ so that $f(\partial I^n)=g(\partial I^n)=x_0$, then I can concatenate them in the following ways as long as $n>1$:
$f\cdot g=\begin{cases} f(2x,y,\cdots) &\text{ for } 0\leq x\leq 1/2\\ g(2x-1,y,\cdots) &\text{ for } 1/2\leq x\leq 1 \end{cases}$
$f \star g= \begin{cases} f(x,2y,\cdots) &\text{ for } 0\leq y\leq 1/2\\ g(x,2y-1,\cdots) &\text{ for } 1/2\leq y\leq 1 \end{cases}$
Now, if I can show that these are homomorphisms for each other, then I'll have that $\pi_n$ is commutative for $n>1$. But this is easy: both $(f\cdot g)\star (h\cdot i)$ and $(f\star h)\cdot (g\star i)$ are the following map:
$(x,y,...)\mapsto\begin{cases} f(2x,2y,\cdots) &\text{ for }(x,y)\in [0,1/2]\times [0,1/2]\\ g(2x-1,2y,\cdots) &\text{ for } (x,y)\in [1/2,1]\times [0,1/2]\\ h(2x,2y-1,\cdots) &\text{ for } (x,y)\in [0,1/2]\times [1/2,1]\\ i(2x-1,2y-1,\cdots) &\text{ for } (x,y)\in [1/2,1]\times [1/2,1] \end{cases}$
Now, by the Eckmann-Hilton argument, these two sorts of composition are the same and are in fact commutative.
The following is relevant to the point made by Qiaochu Yuan.
What is usually called the Eckmann-Hilton argument is actually a special application of the interchange law, whose quite general setting is for double categories, i.e. sets with two distinct category structures such that, and this is the interchange law, each is a morphism for the other. However, even when each of the structures is a groupoid, i.e. all arrows are isomorphisms, the interchange law does not lead to a commutative structure but only that it contains a family of abelian groups. Also the interchange law implies that double groupoids contain the structures known as crossed modules, which occur in homotopy theory and the cohomology of groups. In fact $n$-fold groupoids become more complicated with increasing $n$.
The existence of such double structure, first formulated by C. Ehresmann in the 1960s, raised the question of their potential use in homotopy theory, a question relevant to the history of homotopy groups, and to their abelian nature.
Topologists of the early 20th century were aware that the non commutative nature of the fundamental group was useful in geometry and analysis; that the first homology group was, for a connected space, the fundamental group made abelian; and that the homology groups were defined in all dimensions. Consequently, there was a desire to find higher dimensional versions of the fundamental group, keeping its nonabelian nature.
In 1932, E. Cech submitted to the ICM at Zurich a paper on higher homotopy groups; however, Alexandroff and Hopf, the kings of topology at the time, objected to the fact that they were abelian for $n \geq 2$, and on these grounds persuaded Czech to withdraw his paper, so that only a small paragraph appeared in the ICM Proceedings.
It seems that Hurewicz attended this ICM. In 1935, the first of his notes on homotopy groups was published, and from then the concerns about the abelian nature of higher homotopy groups were regarded as a failure to accept a basic fact of life.
In a publication of 1978 R. Brown and P.J. Higgins defined the homotopy double groupoid $\rho(X,A,x)$ of a pointed pair of spaces, which consisted of homotopy classes of maps of a square $I^2$ into $X$ which mapped the edges to $A$ and the vertices to the base point $x$. This enabled the proof of a 2-dimensional van Kampen theorem, which had as a special case a result of Whitehead on free crossed modules.
This work was in fact inspired by that work by J.H. C. Whitehead who in the 1940s developed a deeper understanding of the nonabelian second relative homotopy group of a pair $(X,A)$, with an operation of $\pi_1(A,x)$, and introduced the notion of crossed module for the structure of the boundary map $\delta: \pi_2(X,A,x) \to \pi_1(A,x)$. He also gave a deep determination of $\pi_2(A \cup \{e^2_\lambda\}_{\lambda \in \Lambda}, A,x)$ as a free crossed $\pi_1(A,x)$-module on the characteristic maps of the $2$-cells.
One advantage of this result is that allows for an expression that we very much want, namely that in the standard representation of a Klein bottle as a square $\sigma$ with boundary we would like to write $$\delta \sigma= a+b-a+b,$$ which is a noncommutative formula, and this is exactly possible in the context of crossed modules, with $\sigma$ as a free generator. In the usual chain complex system we can of course write only
$$\partial \sigma= 2b,$$ thus giving a loss of information.
The interchange law is thus central to the use of higher groupoids in homotopy theory and other areas. There are lots of fun things, with an extra structure of connections to enable such things as the notions of commutative cubes and of rotations.
For more details on maths and history, see the pdf of this book, and also this paper, available there in a special issue of Indagationes Math. in honor of L.E.J. Brouwer.