First: Is there a precise meaning to the term "model for (oo,n)-categories"? A related question, which might actually be the same question, is: what exactly do we want to get out of (oo,n)-category theory for general n? Whatever definition of (oo,n)-categories we use, what are the desired things it should satisfy? What should the main examples be, for general n? I know that [bordism categories][1] should be examples. What else? Actually, aside from the cobordism hypothesis I don't really know what the motivations are for (oo,n)-category theory for general n (or at least n bigger than 1), so I hope that people can say some words about that as well.

Second: Presently, what are the models that we have for (oo,n)-categories? Which models have been proven to be equivalent? Of course, there's already a lot about this on the nLab:

[(oo,1)-categories][2]

[(oo,2)-categories][3]

[(oo,n)-categories][4]

I'm mainly just curious about the current status on (oo,n)-categories for general n. Aside from n-fold complete Segal spaces, are there other definitions/"models"?


  [1]: http://www.math.harvard.edu/~lurie/papers/cobordism.pdf
  [2]: http://ncatlab.org/nlab/show/(infinity,1)-category
  [3]: http://ncatlab.org/nlab/show/(infinity,2)-category
  [4]: http://ncatlab.org/nlab/show/(infinity,n)-category