The history of higher category theory has lead to a wealth of definitions, each built on differing ideas and principles. Until recently there has been very little in the way of machinery to compare them. My ongoing work, joint with Clark Barwick, is changing this.
Our first paper is now available arXiv:1112.0040 (Nov 30,2011 version). In this work we give four axioms that characterize the (∞,1)-category of all (∞,n)-categories. We show that the space of (∞,1)-categories satisfying our axioms is (RP∞)n. This space is connected which shows that these ∞-categories are all equivalent. Finally we show that a host of proposed theories of (∞,n)-categories satisfy our axioms.