2-TQFT are to Frobenius Algebras as ??? are to Hopf Algebras 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.)
 A: Earlier than trying to handle the antipode map, you're dead in the water just trying to see the multiplication and comultiplication in a bialgebra as both corresponding to pairs of pants.  Indeed, in a bialgebra you do not expect that
$$ X \otimes X \quad\overset{\Delta \otimes \operatorname{id}}\longrightarrow\quad X \otimes X \otimes X \quad\overset{\operatorname{id} \otimes m}\longrightarrow\quad X \otimes X $$
should have much to do with the map
$$ X \otimes X \quad \overset m \longrightarrow \quad X \quad \overset\Delta\longrightarrow \quad X \otimes X $$
whereas these are equal in a Frobenius algebra, because of the various ways to decompose the sphere with four punctures into two pairs of pants.
So I think the answer to your question is "no".
A: I'd like to buy some hypotheses.
Hopf algebras are ubiquitous algebraic objects that can be the cohomology ring of an H-space,
the Universal enveloping algebra of a Lie algebra, the group ring of a group, or a crucible
for understanding a solution to the Yang-Baxter equation.  Put on enough additional hypotheses
and you can get several different theorems of the form you are asking for.
A: This is my "geometric freshman explanation":
the problem is to put something instead of the "?" doing the job of $s$ in

...but the composition $\eta\circ\epsilon$ is disconnected, and there is no way to obtain a disconnected manifold starting gluing something to that. :( such a pity.
