The fact that the category of tangles can be "algebraicized" via its braided monoidal structure has played a fundamental role in the theory of quantum and finite-type invariants.
How can this algebraization be generalized to tangles lying in a cylinder over an arbitrary surface?
In this talk I will present one of the possible answers.
Even though the approach may not be the most natural one, the good point is that it is based on a rather explicit description of surface tangles (by some kind of planar diagrams that I call "beak diagrams").