nLab Eckmann-Hilton argument

Statements

In its original form, the Eckmann–Hilton argument shows that a monoid object or group object internal to the category of monoids or Groups is necessarily commutative.

In other words this says that if a set is equipped with a pair of monoid structures, such that one is a homomorphism for the other, then the two structures actually 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 a commutative 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 symmetric 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. In particular, the convex powerset of distributions monad is not commutative.

Uniform states are unique

A related guise of this result is the following: If $C$ is a distributive monoidal category whose unit $I$ is terminal, then any two morphisms $p, q : I \to I + I$ which are symmetric, i.e. invariant under the swap $I + I \cong I+I$, must be equal. From the viewpoint of categorical probability, this says that uniform distributions (if they exist) must be unique.

Every morphism $p : I \to I + \ldots + I$ gives rise to an $n$-ary operation on homsets $\omega_p : C(X,Y)^n \to C(X,Y)$ via

$$\omega_p(f_1,\ldots,f_n) = X \xrightarrow{p \otimes X} (I + \ldots + I) \otimes X \cong X + \ldots + X \xrightarrow{[f_1,\ldots,f_n]} Y$$

The operations $\omega_p$ and $\omega_q$ are idempotent (by terminality of $I$) and commute with each other (by monoidality). Hence if the operations are also symmetric, then they must coincide following the usual Eckmann-Hilton argument.

We recover the connection to algebraic theories if we take $C$ to be the Kleisli category of a monad $T$ which is commutative and affine.

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)