Joyal: A quasicat is a Kan complex iff its homotopy cat is a groupoid.
There is a set of notes on quasicats from a course by Joyal in Barcelona, I think Peter said he would send this at some point.
Toen: Vers une axiomatisation etc. File Toen publ ntopoi.pdf. Defines a “theory of (1-infty)-categories”, and shows that any such theory is Quillen eq to complete Segal spaces. Another example of such a theory is Rezk’s homotopy theory of homotopy theories.
nLab page on Quasi-category