Notes on Atiyah’s Tqft’s

As an example of “categorification”, I presented Atiyah’s axioms [1] for a topological quantum field theory (TQFT) to undergraduates in the University of Chicago’s summer 2002 REU. These notes are written for the use of the participants and to test my own understanding.The aim is just to make very categorically precise what the basic definitions are. Also, there are some details in Atiyah’s original definition that are subsumed in the more recent and by now standard categorical definition, and it seemed worthwhile to present a comparison. I have in mind hitting TQFT’s with infinite loop space machines, which may perhaps provide a little unusual motivation. However, these notes are written to be readable by the students attending the REU, although standard categorical background that was explained in the talks is not repeated here. Needless to say, we have very good students at Chicago!!!