As is well known, the tangle hypothesis of Baez and Dolan proposes that, for suitable definitions, the $n$-category of framed $n$-tangles in $n+k$ dimensions is the free $k$-tuply monoidal $n$-category with duals on one object. For ($n=1,k=2$) and ($n=2,k=2$) these have been proven respectively here and arXiv:math/9811139 (but in the latter case only for unframed tangles). These results are expressed in terms of "classical" notion of $n$-categories. By "classical" n-category, I will mean the usual notion of a structure, with objects, and $1$-morphisms satisfying coherence conditions up to $2$-morphisms, which themselves are required to satisfy coherence conditions, and so forth.
On the other hand, Lurie (arXiv:0905.0465) has proven a version of the hypothesis in the general case, but for a definition of $n$-category that is too abstract for a humble physicist such as myself to do much with.
I have two related questions:
- Are there are any more proofs of special cases of the tangle hypothesis in terms of "classical" $n$-categories, beyond the ones I gave? (Specifically, I am interested in the case $n=2,k=2$, but for framed tangles.)
- Is it possible to extract such results from Lurie's work? (Or in other words, can one translate Lurie's definition of $n$-categories into a "classical" one, at least in low dimensional cases?)
I realize that the "classical" definitions become increasingly cumbersome as you increase $n$, but for $n=2,k=2$ you are dealing with braided monoidal 2-categories, which are still fairly manageable, e.g. see this paper for a general definition of weak braided monoidal 2-categories and a strictification theorem.