Applications of Eckmann–Hilton argument

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.

Solution 1:

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.)

  1. 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.)
  2. A monoidal category is (more or less) a monoid object in the category of categories.
  3. 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!
  4. 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.

Solution 2:

You could look at this answer to a question on 2-categorical structure on the category of topological spaces, and also at this web article on higher dimensional group theory. The title is intended to be slightly wrong, as it should strictly speaking be higher dimensional groupoid theory. But then groupoids are a natural and convenient generalisation of groups.