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

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

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.)?