The question arose this morning during a seminar about HAs.
In a few words: can the equivalence $2-TQFT_k \leftrightarrow Frob_k$ be "modified" in a sensible way to give a similar one between the category $HA$ of Hopf algebras and a suitable "topological" category (I mean: a -even functor- category made 'with' topological objects, hopefully in a sufficiently small neighborhood of $2-TQFT$)? In particular i would like to find a visual analogue for the antipode map $s:H\to H$.
Bad thing is that it takes a while to discover there seem to be no way to define it as an arrow in $Cob(2)$: just try to draw in $Cob(2)$ the diagram
...any sensible choice for $s$ leaves in the manifold one hole more than the minimum. Spending a couple of words about the "sensible choice", it seems to me the only way not to increase the genus of the surface is to take as cobordism a-cap-and-a-cup, namely the [Cob(2)-analogue of the] composition $\eta\circ \epsilon\colon H\to k\to H$ in the former diagram... But I'm not able to characterize it as a Frobenius-Algebra map in any sensible way.
So, help me... (maybe the person I discussed with this morning is here? His website is this.)