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Okay, I'm going to ask a naiive question that surely has an interesting answer. So, a first approximation of defining a (small) weak n-category probably goes something like this. Take a pre-n-category C of all the cells, source and target maps that do the right thing (i.e. are globular), and a composition defined for each r in {0,...,n}.

For C, define a family of coherent sets $(\Sigma\_1, \Sigma\_1, \ldots)$ as a family of sets $\Sigma_r$ of r-cells in C such that

  1. $f : a \rightarrow b \in \Sigma\_r \Rightarrow \exists f' : b \rightarrow a \in \Sigma\_r$
  2. $f, f' : a \rightarrow b \in \Sigma\_r \Rightarrow \exists \alpha : f \rightarrow f' \in \Sigma\_{r+1}$

Now, suppose C admits such a family of coherent sets and all r-cells have associators, uniters, and interchangers in $\Sigma\_{r+1}$, one might be tempted to say C is an $\infty$-category. If for all $r \geq n+1$, $\Sigma\_r$ is only identites, one might say this defines an n-category.

So, the reason I say "one might be tempted to say" is that, if it were that easy, someone much smarter than me would have done it already. :) So, where does the above recipe fail? Or is this definition unsatisfactory because it doesn't express the structure using a finite generating set of commutative diagrams (cf. Mac Lane's coherence etc.)?

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up vote 9 down vote accepted

If I understand correctly what you're getting at, I think the reason this fails is because for n>2, not every diagram of constraints can be expected to commute (even up to higher constraints) in a weak n-category. For example, a braided monoidal category can be regarded as a weak 3-category with one 0-cell and one 1-cell, but then the "double twist" is a constraint isomorphism which is not equal to the identity (also a constraint isomorphism).

One way to get around this is, instead of talking about diagrams of constraints in some particular n-category, to talk about "formal" diagrams of constraints, i.e. diagrams of constraints in a free n-category. I think that when one makes your idea precise using this corrected approach, one will end up with something very similar to Batanin's higher-operadic definition of weak n-category.

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