TITLE: What properties should a good definition of (weak) $n$-category satisfy?
QUESTION [11 upvotes]: My (perhaps inaccurate) impression is there are many competing definitions of the notion of a (weak) $n$-category, none of which are generally accepted.  I've run across several properties such definitions should satisfy, e.g. that weak $n$-groupoids should model homotopy $n$-types, or that the collection of $n$-categories should naturally be an $n+1$-category.  But I haven't found anything like a comprehensive list of conjectural properties a "good" theory of weak $n$-categories should satisfy.

Is there such a list somewhere in the literature?

If not, I'd of course be thrilled with an answer containing such a list.

REPLY [5 votes]: Also read some of the original papers by Baez and Dolan, for example Categorification (in Higher Category Theory, Contemporary Mathematics 230, AMS 1998), where the Tangle Hypothesis is explained among other things. 
It is hoped that a satisfactory definition on n-category would provide a framework for proving the homotopy hypothesis, the tangle hypothesis, and the stabilization conjecture, as discussed at the n-Category Café here. 
I wouldn't say that none of the proposed definitions is generally accepted, but rather that none has emerged as clearly the best definition. In some sense all are accepted as capturing some useful intuition, although some seem easier to work with than others for some specific purpose. Perhaps the most developed notions are the more geometric ones which take advantage of decades of work in homotopy theory.