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 2group, or something like that, by considering the 2category defined by 2link movies. However, I have no recollection of where I saw this (I think it was a dissertation or something).
It also seems that the 2knot 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?



Just as the category of tangles may be thought of as a free braided, monoidal category with duals, the 2category of 2tangles may be thought of as a free braided monoidal 2category with duals, which was proved by Baez and Langford. Part of the program of "categorification" is to work out how to find invariants of 2knots and 4manifolds. The first step may be Khovanov homology. This should hopefully relate to Lauda's categorification of quantum sl2. 


There are many more details that are needed in Eitan's answer. Since weak 2groups are determined by group 3cocycles, one would think that group cocycle conditions are related to 2knots. They are, but the relation is only in relation to quandle cocycles, see for example, Kabaya and Inoue . Khovanov homology gives very weak invariants of closed surfaces in 4space. However, Kevin Walker's answer to my question indicates that Khovanov homology does give a braided monoidal 2category with duals. It also seems likely that some of Jacob Lurie's work might give other neat examples from which invariants of 2tangles can be constructed. In work not yet complete, Alissa Crans, Mohamed Elhamdadi and I are relating strict $2$groups to strict $2$quandles. We are hoping that weak $2$quandles would be parametrized by their $3$cocycles. It seems reasonable that quandle cocycle invariants of classical knots come from a braided monoidal category, and that the associated invariants of knotted surfaces come from braided monoidal $2$categories with duals. My own opinion about skeintype invariants of knotted surfaces is that, if they exist and are not trivial, then they might be coming from the KhovonovRozansky categorifications of MOY $sl_3$ invariants, theories of Soergel bimodules, or work of Ben Webster. I also think that there is a nice intermediate category and $2$category such as this. The category such be described as a Frobenius object in a braided monoidal category. I don't know how to describe the associated $2$category, but it will represent embedded foams in $4$space. Embedded foams are analogous to knotted trivalent graphs. For both of these there are invariants that come from cocycles defined for group families of quandles. We are writing this stuff down now. More generally, it is a good idea to have explicit descriptions of the chain homotopies in a knot homology theory that are induced by the Reidemeister moves. 

