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## Centers, 2-categories, and the Eckmann-Hilton argument

The center $Z(G)$ of a group is an interesting construction: it associates to every group $G$ an abelian group $Z(G)$ in what is certainly a canonical way, but not a functorial way: that is, it doesn’t extend (at least in any obvious way) to a functor $\text{Grp} \to \text{Ab}$ (unlike the abelianization $G/[G, G]$). We might wonder, then, exactly what kind of construction the center is.

Of course, it is actually not hard to come up with a rather general example of a canonical but not functorial construction: in any category $C$ we may associate to an object $c \in C$ its automorphism group $\text{Aut}(c)$ or endomorphism monoid $\text{End}(c)$), and this is a canonical construction which again doesn’t extend in an obvious way to a functor $C \to \text{Grp}$ or $C \to \text{Mon}$. (It merely reflects some special part of the bifunctor $\text{Hom}(-, -)$.)

It turns out that the center can actually be thought of in terms of automorphisms (or endomorphisms), not of a group $G$, but of the identity functor $G \to G$, where $G$ is regarded as a category with one object. This definition generalizes, and the resulting general definition has some interesting specializations. Moreover, an important general property is that the center is always abelian, and this has a very elegant proof.

Some preliminary categorical remarks

A (strict) 2-category is a category enriched over the monoidal category $(\text{Cat}, \times)$. Spelled out, it is a collection of objects, and for each pair of objects $A, B$ a category $\text{Hom}(A, B)$, and for each object $A$ an identity object $\text{id}_A$ in $\text{Hom}(A, A)$, and for each triplet of objects a functor

$\circ : \text{Hom}(A, B) \times \text{Hom}(B, C) \to \text{Hom}(A, C)$

such that all this data satisfies the obvious associativity and identity constraints. Objects in the categories $\text{Hom}(A, B)$ are called 1-morphisms while morphisms in them are called 2-morphisms.

If we think of a category as a collection of dots and arrows between them, then a 2-category is a collection of dots, arrows between them, and 2-cells between the arrows. A 2-morphism can be drawn like this:

(The diagram above may look strange; unfortunately, WordPress doesn’t support xy-pic, so it’s copy-pasted from another document.)

Unlike 1-morphisms, 2-morphisms admit two notions of composition. Vertical composition of 2-morphisms is just the usual composition of morphisms in the category $\text{Hom}(A, B)$ where $A, B$ are two objects. It can be drawn like this:

Horizontal composition of 2-morphisms is the induced action of the functor $\circ : \text{Hom}(A, B) \times \text{Hom}(B, C) \to \text{Hom}(A, C)$ on morphisms. It can be drawn like this:

Vertical and horizontal composition are related by the functoriality of $\circ$, which gives concretely the exchange law. The exchange law says that the two possible ways to compose the 2-morphisms in the diagram

into a single 2-morphism (vertical compositions first, then horizontal, or horizontal compositions first, then vertical) are the same. One can think of the exchange law as a 2-dimensional analogue of the associativity of composition of (1-)morphisms in a (1-)category, which says that the two possible ways to compose the 1-morphisms in the 1-dimensional diagram

$\displaystyle A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D$

into a single 1-morphism are the same.

Example. The primal example of a 2-category is $\text{Cat}$, the 2-category of categories, functors, and natural transformations. More generally, for any monoidal category $V$, we may consider the 2-category of $V$-enriched categories, $V$-enriched functors, and $V$-enriched natural transformations. Here vertical composition is the ordinary composition of functors and horizontal composition is the Godement product.

Example. Since any set may be regarded as a category with only trivial morphisms, any ordinary category (or 1-category) may also be regarded as a 2-category with only trivial 2-morphisms.

Example. Just as a groupoid is a category in which all of the morphisms are invertible, a (strict) 2-groupoid is a 2-category in which all of the 1-morphisms are invertible. Important examples, extending the fundamental groupoids $\Pi_1(X)$, are the fundamental 2-groupoids $\Pi_2(X)$ of topological spaces $X$. Here the objects are the points of $X$, the morphisms are the continuous paths in $X$, and the 2-morphisms are homotopy classes of homotopies between paths. By the homotopy hypothesis for 2-groupoids, these more or less exhaust all examples of 2-groupoids.

Thinking of a homotopy between two paths in $X$ as a continuous map $H : I^2 \to X$, horizontal and vertical composition correspond to two possible ways to combine two copies of the square $I^2$ along a side into another square. (Said another way, $H$ can be regarded either as a homotopy between the paths $H(t, 0), H(t, 1)$ or between the paths $H(0, t), H(1, t)$, and both of these interpretations give rise to a notion of composition.)

Centers

We return now to groups. Let $f, g : G \to H$ be a pair of homomorphisms between two groups (so a pair of functors between the corresponding categories). A natural transformation $\eta$ from $f$ to $g$ associates to the unique object in $G$ a single morphism $h \in H$ such that $hf = gh$, or equivalently such that $hfh^{-1} = g$; moreover, composition of such natural transformations agrees with composition of elements of $H$. Letting $G = H, f = g = \text{id}_G$ it follows that the natural transformations $\text{id}_G \to \text{id}_G$ are precisely the elements $h \in G$ such that $hxh^{-1} = x$ for all $x \in G$, or precisely the elements of $Z(G)$.

If we want to apply this description uniformly to the entire category $\text{Grp}$, we are forced to acknowledge the fact that $\text{Grp}$ is not just a category, but really a 2-category (in fact a sub-2-category of $\text{Cat}$). This motivates the following definition.

Definition: Let $C$ be an object in a 2-category. Its center $Z(C)$ is the monoid of endomorphisms of the identity morphism $\text{id}_C : C \to C$.

Example. Let $C$ be a category (so an object of $\text{Cat}$). Then a natural transformation $\eta : \text{id}_C \to \text{id}_C$ consists of the following data: for every object $x \in C$, an endomorphism $\eta_x : x \to x$ such that for every morphism $f : x \to y$ in $C$ we have

$\displaystyle \eta_y \circ f = f \circ \eta_x$.

Taking $x = y$ we see that each $\eta_x$ must in particular be a central element of $\text{End}(x)$, so $Z(C)$ is commutative as in the group case above.

Recall that a generator of a category is an object $x$ such that any two parallel pair of distinct morphisms $f, g : y \to z$ can be distinguished by morphisms from $x$ in the sense that there exists a morphism $h : x \to y$ such that $fh \neq gh$. In other words, it is an object such that the functor $\text{Hom}(x, -)$ is faithful.

Proposition: Let $x$ denote a generator of a category $C$. Then the natural map

$\eta_x : Z(C) \to Z(\text{End}(x))$

is injective.

Proof. Let $\eta, \eta' \in Z(C)$ and assume that $\eta_x = \eta_x'$. For every morphism $f : x \to y$, we know that

$\eta_y \circ f = f \circ \eta_x = f \circ \eta_x' = \eta_{y}' \circ f$.

Since $x$ is a generator, the above implies that $\eta_y = \eta_y'$ for all $y$, from which it follows that $\eta = \eta'$.

Sub-example. (This is how you know what categorical level we’re working at!) Let $R$ be a (unital, not necessarily commutative) ring and let $C = R\text{-Mod}$ be the category of left $R$-modules. Then $R$, regarded as a left $R$-module, is a generator, so there is a natural injection

$\eta_x : Z(R\text{-Mod}) \to Z(\text{End}_R(R)) \cong Z(R^{op}) \cong Z(R)$.

On the other hand, multiplication by every element of the center $Z(R)$ is readily seen to lift to an element of $Z(R\text{-Mod})$, so the above map is an isomorphism.

It follows that we can recover the center of any ring from the structure of $R\text{-Mod}$ as an $\text{Ab}$-enriched category; that is, the center is a Morita invariant of rings. In particular, any commutative ring can be recovered from the structure of its module category.

Example. In the fundamental 2-groupoid $\Pi_2(X)$ of a topological space $X$, given a point $x \in X$, we have $Z(x) \cong \pi_2(X, x)$, the second homotopy group of $X$ based at $x$. Note that like the center of a category above, $\pi_2(X, x)$ is commutative. We’ll give a unified explanation of this below.

Remark. The center, as a construction associated to an object in a 2-category, can equivalently be regarded as a construction associated to a 2-category with one object. Given such a 2-category, the endomorphisms of the unique object together with the 2-morphisms between them form a (strict) monoidal category, and conversely given a (strict) monoidal category $(M, \otimes)$ we can write down a 2-category with one object with endomorphisms the objects of $M$, 2-morphisms the morphisms of $M$, and composition law $\otimes$. So the center can equivalently be regarded as a construction $\text{End}(I)$ associated to the identity object $I$ in a monoidal category.

The Eckmann-Hilton argument

When we called the center a monoid, we referred implicitly to vertical composition of 2-morphisms. However, horizontal composition also defines a monoid structure on $Z(C)$, and the two are related by the exchange law. This could potentially be quite a complicated structure. However, it turns out to be quite simple.

Proposition: Vertical and horizontal composition agree on $Z(C)$ and are commutative.

Corollary: The second homotopy group $\pi_2(X, x)$ of a pointed topological space is commutative.

Proof (1-dimensional). Letting $\cdot$ denote vertical composition and $\circ$ denote horizontal composition, the exchange law gives

$(\alpha \circ \beta) \cdot (\gamma \circ \delta) = (\alpha \cdot \gamma) \circ (\beta \cdot \delta)$.

Let $e$ denote the identity 2-morphism, which is an identity for both vertical and horizontal composition. Letting $\beta = \gamma = e$,

$\alpha \cdot \delta = (\alpha \circ e) \cdot (e \circ \delta) = (\alpha \cdot e) \circ (e \cdot \delta) = \alpha \circ \delta$

so vertical and horizontal composition agree. Furthermore,

$\alpha \circ \delta = (e \cdot \alpha) \circ (\delta \cdot e) = (e \circ \delta) \cdot (\alpha \circ e) = \delta \cdot \alpha$

so $\cdot$ is commutative.

The above argument is known as the Eckmann-Hilton argument. It can be presented as above using 1-dimensional notation, but this obscures the topological content of the argument, which is better seen using the 2-dimensional notation we used above.

Proof (2-dimensional). Using the notation above, we can write the above proof as follows:

Note that the second and fourth diagrams uniquely define a 2-morphism by the exchange law, so we can avoid explicitly invoking it in the same way that we generally avoid explicitly invoking the associativity of composition in a category. The topological content of the above proof is particularly compelling if we specialize it to a fundamental 2-groupoid.

### 5 Responses

1. Hi Qiaochu, what did you use to create and save the images from xy-pic?

• Nothing fancy. I blew up a .pdf containing the images I wanted (with borders around them) to 200%, print-screened it into mspaint, and saved it as a .png. The borders were so I knew where to cut. There must be a better way to do this, but it works and doesn’t take too much time.

• I ended up do the going through a similar route, except I did not use the borders, which is a great idea. Thanks for the feed back.

2. [...] Comments « Centers, 2-categories, and the Eckmann-Hilton argument [...]

3. […] The basic idea is to observe that a pointed map from to a pointed space is the same thing as a map from the -cube to such that the boundary is sent to . In general, morphisms from the -cube can be glued together along any pair of -dimensional faces provided that the images of those faces match. There are distinguished such gluings coming from gluing together each of the copies of in the product in the usual way that one glues two intervals together. These gluing operations are natural, associative, and have inverses up to homotopy. They give compatible group operations on which, when , make it an abelian group by the Eckmann-Hilton argument. […]