TITLE: Higher order quandle
QUESTION [8 upvotes]: The notion of quandle is known to be closely related to knot theory. The three axioms in the definition of quandle correspond to the Reidemeister moves.
Recently I learned that there are higher analogues of Reidemeister moves. For surface knots. I've heard that seven moves of Roseman determine isotopy classes.
Is there an algebraic structure, which can be regarded as a 2-dimensinal analogue of quandle, corresponding to Roseman moves?

REPLY [7 votes]: The definition of strict 2-quandle and examples thereof has not be written down in a public forum yet. Crans, Elhamdadi, Saito, and I have a notion and examples. I think that Crans spoke about the idea in Riverside, and I will give a talk about it at Knots in Washington next weekend. 
I don't reckon that I will have slides ready before then. If I do, I will post them on my web page. I am considering blackboard talks for the up-coming conference.
It is still conjectural that a weak 2-quandle is an axiomatization of the Roseman moves. I am playing with a couple of diagrammatic schemata to make sure everything works. Also after your question, I started puzzling about the tetrahedral move. This should be a theorem, and right now it appears to be an axiom.