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In short, we are really working with the category of sets. A category is something just as abstract as a set, but a bit more structured. It's not a mere collection of objects; there are also morphisms between objects, in this case the functions between sets.

Some of you might not know the precise definition of a category; let me state it just for completeness. A category consists of a collection of "objects" and a collection of "morphisms". Every morphism f has a "source" object and a "target" object. If the source of f is X and its target is Y, we write f: X → Y. In addition, we have:

1) Given a morphism f: X → Y and a morphism g: Y → Z, there is a morphism fg: X → Z, which we call the "composite" of f and g.

2) Composition is associative: (fg)h = f(gh).

3) For each object X there is a morphism 1 X : X → X, called the "identity" of X. For any f: X → Y we have 1 X f = f 1 Y = f.

That's it.

(Note that we are writing the composite of f: X → Y and g: Y → Z as fg, which is backwards from the usual order. This will make life easier in the long run, though, since fg will mean "first do f, then g".)

Now, there are lots of things one can do with sets, which lead to all sorts of interesting examples of categories, but in a sense the primordial category is Set, the category of sets and functions. (One might try to make this precise, by trying to prove that every category is a subcategory of Set, or something like that. Actually the right way to say how Set is primordial is called the "Yoneda lemma". But to understand this lemma, one needs to understand categories a little bit.)

When we get to thinking about categories a lot, it's natural to think about the "category of all categories". Now just as it's a bit bad to speak of the set of all sets, it's bad to speak of the category of all categories. This is true, not only because Russell's paradox tends to ruin attempts at a consistent theory of the "thing of all things", but because, just as what really counts is the category of all sets, what really counts is the 2-category of all categories.

To understand this, note that there is a very sensible notion of a morphism between categories. It's called a "functor", and a functor F: C → D from a category C to a category D is just something that assigns to each object x of C an object F(x) of D, and to each morphism f of C a morphism F(f) of D, in such a way that "all structure in sight is preserved". More precisely, we want:

1) If f: x → y, then F(f): F(x) → F(y).

2) If fg = h, then F(f)F(g) = F(h).

3) If 1 x is the identity morphism of x, then F(1 x ) is the identity morphism of F(x).

It's good to think of a category as a bunch of dots - objects - and arrows going between them - morphisms. I would draw one for you if I could here. Category theorists love drawing these pictures. In these terms, we can think of the functor F: C → D as putting a little picture of the category C inside the category D. Each dot of C gets drawn as a particular dot in D, and each arrow in C gets drawn as a particular arrow in D. (Two dots or arrows in C can get drawn as the same dot or arrow in D, though.)

In addition, however, there is a very sensible notion of a "2-morphism", that is, a morphism between morphisms between categories! It's called a "natural transformation". The idea is this. Suppose we have two functors F: C → D and G: C → D. We can think of these as giving two pictures of C inside D. So for example, if we have any object x in C, we get two objects in D, F(x) and G(x). A "natural transformation" is then a gadget that draws an arrow from each dot like F(x) to the dot like G(x). In other words, for each x, the natural transformation T gives a morphism T x : F(x) → G(x). But we want a kind of compatibility to occur: if we have a morphism f: x → y in C, we want

F(f) F(x) -----> F(y) | | T x | |T y v v G(x) -----> G(y) G(f)

x

y

This must seem very boring to the people who understand it and very mystifying to those who don't. I'll need to explain it more later. For now, let me just say a bit about what's going on. Sets are "zero-dimensional" in that they only consist of objects, or "dots". There is no way to "go from one dot to another" within a set. Nonetheless, we can go from one set to another using a function. So the category of all sets is "one-dimensional": it has both objects or "dots" and morphisms or "arrows between dots". In general, categories are "one-dimensional" in this sense. But this in turn makes the collection of all categories into a "two-dimensional" structure, a 2-category having objects, morphisms between objects, and 2-morphisms between morphisms.

This process never stops. The collection of all n-categories is an (n+1)-category, a thing with objects, morphisms, 2-morphisms, and so on all the way up to n-morphisms. To study sets carefully we need categories, to study categories well we need 2-categories, to study 2-categories well we need 3-categories, and so on... so "higher- dimensional algebra", as this subject is called, is automatically generated in a recursive process starting with a careful study of set theory.

If you want to show off, you can call the 2-category of all categories Cat, and more generally, you can call the (n+1)-category of all n-categories nCat. nCat is the primordial example of an (n+1)-category!

Now, just as you might wonder what comes after 0,1,2,3,..., you might wonder what comes after all these n-categories. The answer is "ω-categories".

What comes after these? Well, let us leave that for another time. I'd rather conclude by mentioning the part that's the most fascinating to me as a mathematical physicist. Namely, the various dimensions of category turn out to correspond in a very beautiful - but still incompletely understood - way to the various dimensions of spacetime. In other words, the study of physics in imaginary 2-dimensional spacetimes uses lots of 2-categories, the study of physics in a 3d spacetimes uses 3-categories, the study of physics in 4d spacetimes appears to use 4-categories, and so on. It's very surprising at first that something so simple and abstract as the process of starting with sets and recursively being led to study the (n+1)-category of all n-categories could be related to the dimensionality of spacetime. In particular, what could possibly be special about 4 dimensions?

Well, it turns out that there are some special things about 4 dimensions. But more on that later.

To continue reading the `Tale of n-Categories', click here.

Addendum: Long after writing the above, I just saw an interesting article on chirality in biology:

2) N. Hirokawa, Y. Tanaka, Y. Okada and S. Takeda, Nodal flow and the generation of left-right asymmetry, Cell 125 1 (2006), 33-45.

It reports on detailed studies of how left-right asymmetry first shows in the development of animal embryos. It turns out this asymmetry is linked to certain genes with names like Lefty-1, Lefty-2, Nodal and Pitx2. About half of the people with a genetic disorder called Kartagener's Syndrome have their organs in the reversed orientation. These people also have immotile sperm and defective cilia in their airway. This suggests that the genes controlling left-right asymmetry also affect the development of cilia! And the link has recently been understood...

The first visible sign of left-right asymmetry in mammal embryos is the formation of a structure called the "ventral node" after the front-back (dorsal-ventral) and top-bottom (anterior-posterior) symmetries have been broken. This node is a small bump on the front of the embryo.

It has recently been found that cilia on this bump wiggle in a way that makes the fluid the embryo is floating in flow towards the left. It seems to be this leftward flow that generates many of the more fancy left-right asymmetries that come later.

How do these cilia generate a leftward flow? It seems they spin around clockwise, and are tilted in such a way that they make a leftward swing when they are near the surface of the embryo, and a rightward swing when they are far away. This manages to do the job... the article discusses the hydrodynamics involved.

I guess now the question becomes: why do these cilia spin clockwise?

© 1996 John Baez

baez@math.removethis.ucr.andthis.edu

