Higher dimensional categories: an illustrated guide book.

With Aaron Lauda, 2004.

Available here.

This informal guide grew out of notes we made when I was teaching Aaron about n-categories in preparation for the IMA workshop on n-categories in June 2004. We made it available before the workshop to help people, and revised it afterwards. We aimed to make it informal and friendly, with tons of pictures to help the intuition along. There are even two 3D figures to cut out and make!

Abstract:

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.