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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.

Is there such a list somewhere in the literature?

If not, I'd of course be thrilled with an answer containing such a list.

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The first place I'd check is ncatlab.org/nlab/show/n-category . –  Qiaochu Yuan Dec 19 '10 at 23:32
Hmm...that's sort of orthogonal to what I'm looking for. ncatlab has lots of good stuff, but I didn't see the sort of philosophical discussion I was hoping for. –  Daniel Litt Dec 20 '10 at 2:31
There are plenty of links to papers, some of which should have that kind of discussion. –  Qiaochu Yuan Dec 20 '10 at 4:17
Oh of course, there's plenty of stuff along these lines, and I've been noting it down as I go through these papers (as well as lots of Baez's stuff); if no one posts a reasonable list, I'll post one below as CW. I was hoping someone had already collected such a list, or even better, that there was some interesting conjecture which was true conditional on the existence of n-categories with properties 1), 2), 3), ... –  Daniel Litt Dec 20 '10 at 5:58
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4 Answers

up vote 7 down vote accepted

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.

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 $k>n$.

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+1; this is excellent. –  Daniel Litt Dec 20 '10 at 8:44
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To complement Andy's answer, there is also Tom Leinster's A survey of definitions of n-category, which has a good 'further reading' section. He also emphasises that there are many other interesting objects in higher category theory, and my guess is that these provide examples for, and non-trivial extensions of, $n$-categories.

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Also read some of the original papers by Baez and Dolan, for example Categorification (in Higher Category Theory, Contemporary Mathematics 230, AMS 1998), where the Tangle Hypothesis is explained among other things.

It is hoped that a satisfactory definition on n-category would provide a framework for proving the homotopy hypothesis, the tangle hypothesis, and the stabilization conjecture, as discussed at the n-Category Café here.

I wouldn't say that none of the proposed definitions is generally accepted, but rather that none has emerged as clearly the best definition. In some sense all are accepted as capturing some useful intuition, although some seem easier to work with than others for some specific purpose. Perhaps the most developed notions are the more geometric ones which take advantage of decades of work in homotopy theory.

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I'm far from an expert on this subject, but the book "Higher-Dimensional Categories: an illustrated guide book" by Cheng and Lauda (available here) has a very nice description of the things that various competing notions of n-categories are trying to generalize and the motivations behind them. It's not quite a cut-and-dried "list of properties", but my impression is that the philosophies behind the various definitions are too complicated to just make a single list.

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