Date: Thu, 11 Nov 93 11:18:42 GMT From: jrk@sys.uea.ac.uk (Richard Kennaway)

I have a rather basic but vague question.

Categories, functors, and natural transformations seem to form the first three elements of a series which could be continued indefinitely. Why is it that these three suffice, and that further members of the sequence are almost never required?

They don't, and they are. Lo, before the forties, everyone thought they knew what "natural" was and thought not of defining it. Then came the definition [E-ML] in terms of "categories". Categories themselves were seen as unnatural and unnecessary for many years; and still are by some. As you have indicated, "2-categories" are now seen as quite natural by many. To some of us, n-categories are as natural as n-simplexes and n-cubes.

Similarly, there are (1-)categories and 2-categories, and further members of this sequence can be defined, yet they are rarely needed. I have once seen 4-categories referred to, but only once.

Is there some intuitive explanation for this?

There may be a mathematical reason for feeling there is a barrier at n = 3. It is easy to define n-categories and there are interesting examples of these. But structural examples often form something less; let's call them weak n-categories: the r-th composition is only associative up to coherent (r-1)-equivalence etc. You may have met weak 2-categories which are Benabou's "bicategories". Now every bicategory is equivalent (in the approp sense) to a 2-category. Yet, not all weak 3-categories (Gordon-Power-St have called them tricategories) are equiv to 3-categories. Fear of the unknown is natural; at the 3-level and above we are forced into more unfamiliar and exotic phenomena. Exotic, yes, but not unnatural. This is where cubes and braids lurk, and who would call them unnatural? Regards, Ross ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++