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.)
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
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.