TITLE: Applications of Eckmann–Hilton argument
QUESTION [14 upvotes]: I am looking for applications of the Eckmann–Hilton argument. I found one application in Algebraic Topology where we show that the fundamental group is abelian in case of a topological group. Thank You.

REPLY [36 votes]: There are tons of applications! There are also many useful reformulations. The following will be more or less in increasing level of sophistication. But first, a silly picture:

I've written a blog post which expands a lot upon this answer, if you're interested.

As you mentioned, it can be used to prove that the fundamental group of a topological group is abelian. More generally, the fundamental group of an $H$-space is commutative, for exactly the same reasons.
It also shows that the higher homotopy groups of any space are abelian. This is a "particular case" of the preceding point, because $\pi_n(X) \cong \pi_1(\Omega^{n-1}X)$ and $\Omega^{n-1}X$ (the iterated loop space) is an $H$-space (for $n-1 \ge 1 \iff n \ge 2$).
A monoid object in the category of monoids is a commutative monoid. This is literally the statement of the Eckmann–Hilton argument if you think about it, but it's a very useful reformulation, because it helps in generalizing it to other contexts. Similarly, a group object in the category of groups is an abelian group.
If $X$ is a cogroup object and $G$ is a group object in a category $\mathsf{C}$, then $\hom_\mathsf{C}(X,G)$ is an abelian group. This is a generalization of the first point: $S^1$ is a cogroup object via the pinching map, and $\pi_1(G) \cong \hom_{\mathsf{Ho}(\mathsf{Top}_*)}(S^1, G)$ for a pointed space $G$. So if $G$ is a topological group, you immediately get that $\pi_1(G)$ is commutative.
The center $Z(\mathsf{C}$) of a category $\mathsf{C}$ is defined to be the monoid of natural transformations $\operatorname{id}_\mathsf{C} \to \operatorname{id}_\mathsf{C}$. If $\mathsf{C}$ is a category with one object, AKA a monoid, it's straightforward to check that this coincides with the notion of center of a monoid: $Z(M) = \{ x \in M : xy = yx \, \forall y \}$. And just like the center of a monoid is commutative, the center of a category is commutative. This follows from the Eckmann–Hilton argument: elements of $Z(\mathsf{C})$ can either be composed horizontally or vertically, and the two are compatible.
Another notion of "center of a category" exists: when $(\mathsf{C}, \otimes, 1)$ is a monoidal category, you can consider the monoid $Z(\mathsf{C}, \otimes) = \operatorname{End}(1)$ of endomorphisms of the unit object $1$. This has two multiplications: the usual composition, and the tensor product (if $f,g : 1 \to 1$, then $f \otimes g : 1 \cong 1 \otimes 1 \xrightarrow{f \otimes g} 1 \otimes 1 \cong 1$). With a bit of work you can show that this second law is well-defined, and compatible with composition, thus by the Eckmann–Hilton argument $\operatorname{End}(1)$ is commutative.
A generalization of the two notions of the center of a category I talked about is the following. Let $\mathsf{C}$ be a $2$-category, then the endomorphisms of the identity $\operatorname{End}(\operatorname{id}_x)$ of any object $x \in \mathsf{C}$ has two multiplications, vertical and horizontal composition. Thus the two laws are the same and it's a commutative monoid.

To recover the center $Z(\mathsf{C})$, consider the $2$-category $\mathsf{Cat}$ of categories and let $\mathsf{C} \in \mathsf{Cat}$, then $Z(\mathsf{C}) = \operatorname{End}(\operatorname{id}_\mathsf{C})$.
To recover the center $Z(\mathsf{C}, \otimes)$, consider the monoidal category $(\mathsf{C}, \otimes)$ as a $2$-category $\Sigma \mathsf{C}$ with a single object $*$ (this is the "suspension"), then $Z(\mathsf{C}, \otimes) = \operatorname{End}(\operatorname{id}_*)$.

(The following is more advanced, and purposefully vague to not get bogged down in technical details.)

A category is a category. (I didn't manage to get Markdown to start the list at 0, but consider this to be the zeroth case.)
A monoidal category is (more or less) a monoid object in the category of categories.
A $2$-fold monoid object in the category of categories (ie. a category $\mathsf{C}$ with two bifunctors $\otimes_1, \otimes_2 : \mathsf{C}^2 \to \mathsf{C}$ that are "compatible" in the Eckmann–Hilton sense) is a braided monoidal category, not a symmetric monoidal category, as one might expect!
But a $3$-fold monoidal category is a symmetric monoidal category, and it stabilizes: a $k$-fold monoidal category is symmetric for $k \ge 3$.

This is a higher version of the Eckmann–Hilton argument. In Higher category theory you have similar generalizations, by considering $k$-fold monoidal $n$-categories. You should look at the table 21 in Baez & Dolan's article cited below.
An $E_n$-algebra (ie. an algebra over the little disks operad) is, in some sense, a (strongly) homotopy-associative algebra, which is homotopy commutative "up to level $n$". One can make an alternative definition in the spirit of Eckmann–Hilton: consider a symmetric monoidal $(\infty, 1)$-category $\mathsf{C}$. Then an $E_n$-algebra in $\mathsf{C}$ can be defined as an object $A$ equipped with $n$ compatible multiplications (cf. Lurie below).
The link with braided monoidal categories can also be made here. The nerve of an $n$-fold monoidal category is an $E_n$-algebra (cf. Balteanu–Fiedorowicz–Schwänzl–Vogt below).


Useful references, and also interesting articles related to what I said at the end:

Eckmann–Hilton argument at Wikipedia.
These two blog posts by John Baez: number 1, number 2. The second one has a very nice graphical representation of the argument.
John C. Baez and James Dolan. “Higher-dimensional algebra and topological
quantum field theory”. In: J. Math. Phys. 36.11 (1995), pp. 6073–6105. ɪssɴ: 0022-2488. ᴅᴏɪ: 10.1063/1.531236. arXiv: q-alg/9503002 [math.QA]. MR1355899.
C. Balteanu, Z. Fiedorowicz, R. Schwänzl, and R. Vogt. “Iterated monoidal categories”. In: Adv. Math. 176.2 (2003),
pp. 277–349. ɪssɴ: 0001-8708. ᴅᴏɪ: 10.1016/S0001-8708(03)00065-3. arXiv: math/9808082 [math.AT]. MR1982884.
Jacob Lurie. “On the classification of topological field theories”. In: Current developments in mathematics, 2008. Int. Press, Somerville, MA, 2009, pp. 129–280. arXiv: 0905.0465 [math.CT]. MR2555928.