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.



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> Is there such a list somewhere in the literature?
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If not, I'd of course be thrilled with an answer containing such a list.



What properties should a good definition of (weak) $n$-category satisfy?
I've just heard a talk by Julia Bergner on the subject, reporting on her ongoing work with Charles Rezk.
Here's the list of properties that they are trying to establish for their model of $(\infty,n)$-categories. 


* For each $n>0$, the category of $(\infty,n)$-categories is a cartesian model category (i.e. it's a model category, and it's compatible with cartesian product).
* The category of $(\infty,0)$-categories is Quillen equivalent to the category of topological spaces.
* For any $n\ge 0$, the category of $(\infty,n+1)$-categories is Quillen equivalent to the category of categories enriched over $(\infty,n)$-categories.


Clearly, not every model for $(\infty,n)$-categories needs to comply to the above list of properties. But it's also quite clear that *if* a model for $(\infty,n)$-categories complies to the above requirements, then it's a good model.




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And just for the record, an $(\infty,n)$-category is supposed to capture the idea of an $\infty$-category, all of whose $k$-moorphisms are invertible for $k>n$.


