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 should be examples. What else? Actually, aside from the cobordism hypothesis and other TQFT-y things 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. (For n=1, there seems to be a lot of motivation, see for example thisthis or thisthis or thisthis or thisthis or ...)
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:
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"?
