The center of a group is an interesting construction: it associates to every group an abelian group 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 (unlike the abelianization ). 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 we may associate to an object its automorphism group or endomorphism monoid ), and this is a canonical construction which again doesn’t extend in an obvious way to a functor or . (It merely reflects some special part of the bifunctor .)

It turns out that the center can actually be thought of in terms of automorphisms (or endomorphisms), not of a group , but of the *identity functor* , where 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 . Spelled out, it is a collection of objects, and for each pair of objects a category , and for each object an identity object in , and for each triplet of objects a functor

such that all this data satisfies the obvious associativity and identity constraints. Objects in the categories 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 where are two objects. It can be drawn like this:

Horizontal composition of 2-morphisms is the induced action of the functor on morphisms. It can be drawn like this:

Vertical and horizontal composition are related by the functoriality of , 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

into a single 1-morphism are the same.

*Example.* The primal example of a 2-category is , the 2-category of categories, functors, and natural transformations. More generally, for any monoidal category , we may consider the 2-category of -enriched categories, -enriched functors, and -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 , are the **fundamental 2-groupoids** of topological spaces . Here the objects are the points of , the morphisms are the continuous paths in , 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 as a continuous map , horizontal and vertical composition correspond to two possible ways to combine two copies of the square along a side into another square. (Said another way, can be regarded either as a homotopy between the paths or between the paths , and both of these interpretations give rise to a notion of composition.)

**Centers**

We return now to groups. Let be a pair of homomorphisms between two groups (so a pair of functors between the corresponding categories). A natural transformation from to associates to the unique object in a single morphism such that , or equivalently such that ; moreover, composition of such natural transformations agrees with composition of elements of . Letting it follows that the natural transformations are precisely the elements such that for all , or precisely the elements of .

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

**Definition:** Let be an object in a 2-category. Its **center** is the monoid of endomorphisms of the identity morphism .

*Example.* Let be a category (so an object of ). Then a natural transformation consists of the following data: for every object , an endomorphism such that for every morphism in we have

.

Taking we see that each must in particular be a central element of , so is commutative as in the group case above.

Recall that a **generator** of a category is an object such that any two parallel pair of distinct morphisms can be distinguished by morphisms from in the sense that there exists a morphism such that . In other words, it is an object such that the functor is faithful.

**Proposition:** Let denote a generator of a category . Then the natural map

is injective.

*Proof.* Let and assume that . For every morphism , we know that

.

Since is a generator, the above implies that for all , from which it follows that .

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

.

On the other hand, multiplication by every element of the center is readily seen to lift to an element of , so the above map is an isomorphism.

It follows that we can recover the center of any ring from the structure of as an -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 of a topological space , given a point , we have , the second homotopy group of based at . Note that like the center of a category above, 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 we can write down a 2-category with one object with endomorphisms the objects of , 2-morphisms the morphisms of , and composition law . So the center can equivalently be regarded as a construction associated to the identity object 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 , 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 and are commutative.

**Corollary:** The second homotopy group of a pointed topological space is commutative.

*Proof (1-dimensional).* Letting denote vertical composition and denote horizontal composition, the exchange law gives

.

Let denote the identity 2-morphism, which is an identity for both vertical and horizontal composition. Letting ,

so vertical and horizontal composition agree. Furthermore,

so 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.

on February 9, 2012 at 9:14 am |Luis GuzmanHi Qiaochu, what did you use to create and save the images from xy-pic?

on February 9, 2012 at 10:23 am |Qiaochu YuanNothing 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.

on February 10, 2012 at 7:17 am |Luis GuzmanI 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.

on February 16, 2012 at 7:35 pm |Morita equivalence and the bicategory of bimodules « Annoying Precision[…] Comments « Centers, 2-categories, and the Eckmann-Hilton argument […]

on September 8, 2013 at 9:52 am |The homotopy groups are only groups | Annoying Precision[…] 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. […]

on February 23, 2014 at 6:22 am |Ronnie BrownOne should also mention in this context the notion of groupoid. The Eckmann-Hilton argument is one way of showing higher homotopy groups are abelian, a fact which led Alexandroff and Hopf at the Zurich ICM in 1932 to discourage E. Cech from continuing to work in this area.

Different possibilities arise from using goroupoids, since group objects in groupoids, or, equivalently, , groupoid objects groups, are equivalent to crossed modules, which are in some sense “more nonabelian” than groups, and should be thought of as 2-dimensional groups, since they model weak, pointed, homotopy 2-types. There is much more on this in my expository article

R. Brown `Groupoids and crossed objects in algebraic topology’,

Homology, homotopy and applications, 1 (1999) 1-78.(electronic)

which also contains a matrix version of the Eckmann-Hilton argument,

This also explains about the use of certain kinds of double groupoids, which have advantages over the commonly used 2-groupoids. A homotopy strict double groupoid is defined for a pointed pair of spaces, $(X,A,x)$ and is “equivalent” to the second relative homotopy group of the pair, considered as a crossed module. One takes homotopy classes of maps of a square to $X$ which take the edges of the square to $A$ and the vertices to the base point. I find these much easier to use than the globular 2-groupoids, and

I get confused about the use of the term “fundamental 2-groupoid” of a space, since it is not a strict structure, even in dimension 1!

on October 26, 2015 at 8:17 am |Dualizable objects (and morphisms) | Annoying Precision[…] should, but will not, include diagrams, so pretend that I’ve inserted some string diagrams or globular diagrams where […]

on November 9, 2015 at 9:37 pm |Group actions on categories | Annoying Precision[…] Then each is a natural isomorphism , or equivalently an element of the group of units of the center , and the compatibility condition […]

on November 13, 2015 at 6:10 pm |Projective representations give categorical representations | Annoying Precision[…] Proof. First, we observe that a -linear Morita equivalence induces a -linear isomorphism on centers […]