DRAFT VERSION, PREPARED FOR IMA WORKSHOP
Revised afterwards: August 2004
This work gives an explanatory introduction to various definitions of higher-dimensional category. The emphasis is on ideas rather than formalities; the aim is to shed light on the formalities by emphasising the intuitions that lead there. To aid this, the tone is informal and there are copious pictures. The definitions treated are those of Baez-Dolan, Batanin, Joyal, Leinster, May, Penon, Simpson, Street, Tamsamani and Trimble.
This is a draft version prepared for the IMA workshop on n-categories, June 2004.
We can regard the many definitions of n-category as an obstacle or a treasure-trove. In fact, it should always be helpful to have many different views of a structure --- each viewpoint gives us greater scope for understanding. However, it also gives us a greater scope for being baffled.
The aim of this work is to promote understanding. We too have been thoroughly baffled, but have maintained a belief that each definition can look natural if viewed from the appropriate angle. Our general mission, then, is demystification. We will take each definition in turn and find a way of making everything seem as intuitive as possible. Of course, everyone arrives with different intuitions and we also arrived with our own, but we have tried to find out what the appropriate intuitions are that make each idea seem natural. This has involved shifting our mind-set around as we move between different theories, a sort of mental gymnastics that we feel is an essential warm-up for the exercise of understanding.
This work grew out of a very real situation: the second named auther asked the first for an introduction to the various definitions of n-category, in advance of the IMA n-categories workshop in June 2004. A short discussion would not suffice, and we embarked on a journey of discovery in a series of about twenty afternoons. We made detailed notes of everything explained and, motivated by the impending workshop, expanded them into the present form. Our aim here is just as it was when we first sat down with our coffees: to shed as much light on the definitions as we can. These notes are by-and-large a record of our discussions and as such we lay no claims to completeness; we have simply included everything we found helpful along the way. We have chosen intuition over rigour wherever a choice seems to be required. This often means "waffle" over "concision" as we have aimed towards readable prose rather than elegant mathematical exposition that demands hours (or days) of reading and re-reading, sentence by sentence.