nLab Eckmann-Hilton argument

Statements

In its usual form, the Eckmann–Hilton argument shows that a monoid object or group object internal to the category of monoids or Groups is commutative. In other terms, if a set is equipped with two monoid structures, such that one is a homomorphism for the other, then the two structures coincide and the resulting monoid is commutative.

From the nPOV, we may want to think of the statement in this way:

Proposition

Let $C$ be a 2-category and $x \in C$ an object. Write $Id_x$ for the identity morphism of $X$ and $End(Id_x)$ for the set of endo-2-morphisms on $X$. Then:

• horizontal composition and vertical composition define the same monoid object structure on $End(Id_x)$;

• this is an abelian monoid.

On the face of it, this is a special case of the general situation, although in fact every case is an example for appropriate $C$.

A more general version is this: If a set is equipped with two binary operations with identity elements, as long as they commute with each other in the sense that one is (with respect to the other) a homomorphism of sets with binary operations, then everything else follows:

1. the other is also a homomorphism with respect to the first;
2. the identities are the same;
3. the operations are the same;
4. the operation is commutative;
5. the operation is associative.

This can also be internalised in any monoidal category.

Proofs

String diagrams allow an almost trivial proof. Since there is only one object, and the only 1-morphism is the identity, the diagram of $a \circ b$ (vertical composition) is simply two dots labelled $a, b$ arranged vertically. This diagram can be morphed continuously to a horizontal arrangement, which is the diagram for $a * b$ (horizontal composition). This is then morphed to $a$ below $b$, which is the diagram for $b \circ a$.

A pasting diagram-proof of is depicted in Cheng below. Here we prove the $6$-element general form in $Set$.

Proof

The basic equation that we have (that one operation $*$ is a homomorphism with respect to another operation $\circ$) is

$$(a \circ b) * (c \circ d) = (a * c) \circ (b * d) .$$

In $End(Id_x)$, this is the exchange law.

We prove the list of results from above in order:

1. Simply read the basic equation backwards to see that $\circ$ is a homomorphism with respect to $*$.

2. Then

   $$1_\star = 1_\star * 1_\star = (1_\star \circ 1_\circ) * (1_\circ \circ 1_\star) = (1_\star * 1_\circ) \circ (1_\circ * 1_\star) = 1_\circ \circ 1_\circ = 1_\circ ,$$

   so the identities are the same; we will now write this identity simply as $1$.

3. Now

   $$a * b = (a \circ 1) * (1 \circ b) = (a * 1) \circ (1 * b) = a \circ b ,$$

   so the operations are the same; we will write them both with concatenation.

4. Then

   $$a b = (1 a) (b 1) = (1 b) (a 1) = b a ,$$

   so this operation is commutative.

5. Finally,

   $$(a b) c = (a b) (1 c) = (a 1) (b c) = a (b c) ,$$

   so the operation is associative.

If you start with a monoid object in $Mon$, then only (4&5) need to be shown; the others are part of the hypothesis. This classic form of the Eckmann–Hilton argument may be combined into a single calculation:

$$a * b = (a \circ 1) * (1 \circ b) = (a * 1) \circ (1 * b) = a \circ b = (1 * a) \circ (b * 1) = (1 * b) \circ (a * 1) = b * a ,$$

where the desired results involve the first, middle, and last expressions.

Corollaries

A $2$-tuply monoidal $0$-category, if defined as a pointed simply connected bicategory, is also the same as an abelian monoid.

A $2$-tuply monoidal $1$-category, if defined as a pointed simply connected tricategory, is the same as a braided monoidal category.

Every homotopy group $\pi_n$ for $n \geq 2$ is abelian.

Variation

There are variations on the Eckmann-Hilton argument that do not assume units. For example, if a set is equipped with two symmetric $(a * b = b * a)$ and idempotent $(a * a = a)$ binary operations that commute with each other, then the operations coincide.

$$a * b = (a * b) + (a * b) = (a * b) + (b * a) = (a + b) * (b + a) = (a + b) * (a + b) = a + b.$$

For example, we might consider the algebraic theory of convex spaces, and the algebraic theory of semilattices. These theories both contain symmetric idempotent operations: in the theory of convex spaces, take the operation $a * b=c_{0.5}(a,b)$. Thus there can be no commutative algebraic theory that includes these two theories without conflating them. Furthermore, in any conflated theory, all the dyadic rationals are the same, e.g.

$$c_{0.25}(a,b)=c_{0.5}(c_{0.5}(a,b),b)=(a*b)*b=a*(b*b)=a*b=c_{0.5}(a,b).$$

This is relevant in computer science because probability is modelled by the free convex spaces monad, and non-determinism is modelled by the free semilattice monad. These monads are both commutative monads, but there can be no commutative monad that contains both these monads non-degenerately.

Related concepts

• internalization

• group object

• H-space

• exchange law

References

Due to

• Beno Eckmann, Peter Hilton, Theorem 1.12 in: Structure maps in group theory, Fundamenta Mathematicae 50 (1961), 207-221 (doi:10.4064/fm-50-2-207-221)

reviewed (somewhat imperfectly) in:

• Beno Eckmann, Peter Hilton, Theorem 5.4.2 in: Group-like structures in general categories I multiplications and comultiplications, Math. Ann. 145, 227–255 (1962) (doi:10.1007/BF01451367)

Expositions:

• The Catsters

  Eckmann-Hilton 1 (video)

  Eckmann-Hilton 2 (video)

• Eugenia Cheng, (picture)

• Andrew Stacey, animation (mpeg)

Formulation in homotopy type theory:

• Univalent Foundations Project, Thm. 2.1.6 in: Homotopy Type Theory – Univalent Foundations of Mathematics 2013 (web, pdf)

• Kristina Sojakova, Syllepsis in Homotopy Type Theory, 2021 (pdf)

Generalization to higher algebra/higher category theory via operads/(∞,1)-operads:

• Michael Batanin, The Eckmann-Hilton argument and higher operads , Adv. Math. 217 (2008), no. 1, 334–385; (arXiv:math.CT/0207281)

• Tomer Schlank, Lior Yanovski, The ∞-Categorical Eckmann-Hilton Argument, Algebr. Geom. Topol. 19 (2019) 3119-3170 (arXiv:1808.06006, doi:10.2140/agt.2019.19.3119)