Turaev developed the notion of a quantum group by considering the category of tangles (thought of with objects as collections of 2$n$ points and with morphisms being braids between them with cups and caps) and considering the algebraic version of this.  

I see to recall that someone has developed a notion of a quantum 2-group, or something like that, by considering the 2-category defined by 2-link movies. However, I have no recollection of where I saw this (I think it was a dissertation or something).
It also seems that the 2-knot category, considered as a standard category, should define some family of algebraic objects - has this been studied, and if so where should I look for information on this?



2-tangles and quantum groups and 2-groups
Just as the category of tangles may be thought of as a free braided, monoidal category with duals, the 2-category of 2-tangles may be thought of as a free braided monoidal 2-category with duals, which was proved by [Baez and Langford](http://arxiv.org/abs/q-alg/9703033). Part of the program of "categorification" is to work out how to find invariants of 2-knots and 4-manifolds. The first step may be [Khovanov homology](http://arxiv.org/abs/math/9908171). This should hopefully relate to [Lauda's categorification of quantum sl2](http://arxiv.org/abs/0803.3652).


