Why are $n$-fold complete segal spaces or $(\infty, n)$-categories (which I'm unsure of how to distinguish from omega-categories) important for $n \geq 3$? Why are they "badly behaved" for $n \geq 3$? (Lurie refers to them this way in his thesis).


Also, I'm particularly interested to connections between $n$-fold complete segal spaces with regards to a question asked recently about "same" proofs. Is a 2-fold complete segal space sufficient in this particular arena?


(Please tell me if this question is ill-posed, I'm just currently learning category theory.)



omega-categories and n-fold complete segal spaces
$n$-fold complete Segal spaces are one *model* for $(\infty,n)$-categories; there are other models. More precisely, they are supposed to be a model for *weak* $(\infty,n)$-categories.


The distinction that I think you are asking about is between *weak* and *strict*. Strict $n$-categories can be easily defined by a recursive definition: a strict $n$-category is just a category enriched over strict $(n-1)$-categories. A strict 1-category is just a plain-old category. Though easy to define, strict $n$-categories don't seem to capture the things people want an $n$-category to capture. 


One such feature is that strict $n$-categories don't satisfy the "homotopy hypothesis", which says that an $n$-groupoid ($=n$-category in which all morphisms are in some sense invertible) should model homotopy $n$-types ($=$ spaces whose homotopy groups vanish above dimension $n$).
In fact, this failure only occurs for $n \geq 3$; I believe this is the type of bad behavior Lurie refers to. Another failure of strict $n$-categories happens when you try to talk about higher monoidal structures.


If you haven't already, take a look at the papers by Baez-Dolan on arxiv, which discuss a lot of these issues.


