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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 >= 3? Why are they "badly behaved" for n >= 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)

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What do you mean by "same" proofs? And could you say where exactly one can find Lurie's statement? – Peter Arndt Nov 2 '09 at 15:36
I believe he is refering to question 3776: – Charles Rezk Nov 2 '09 at 15:58
Thanks Charles, I was gone for a while so I couldn't give an answer – Michael Hoffman Nov 2 '09 at 16:02
Lurie's comment can be found here (Unfortunately it's kind of long) – Michael Hoffman Nov 2 '09 at 16:05
Ah, he is referring to "strict", not all n-fold segal spaces – Michael Hoffman Nov 2 '09 at 16:07
up vote 8 down vote accepted

n-fold complete Segal spaces are one model for (∞,n)-categories; there are other models. More precisely, they are supposed to be a model for weak (∞,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>=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.

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