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2 fixed typo

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.

1

n-fold complete Segal spaces are one model for (∞,n)-categories; there are 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.