For several combinatorial objects (for example Dyck paths, subsets, integer compositions...) there is some nice "categorification" of them via quivers. That is there is a bijection from those combinatorial objects to quiver algebras that have nice properties.
I wonder what such a nice categorification might be for the maps $f : \{1,...,n \} \rightarrow \{1,...,n \}$.
There is one obvious thing to do, but it does not lead to very nice quiver aglebras. namely let $Q_f$ be the graph of $f$ and $A_f=kQ_f /J^2$, the radical square zero algebra with quiver $Q_f$. This algebra is not nice, since the quiver $Q_f$ is in general disconencted and it has no interesting homological properties except in some small cases.
Question: Is there a nice categorification of maps via connected (!) quiver algebras $A$ such that that the algbras $A$ are quasi-frobenius 2 algebras (meaning all indecomposable projective modules have simple socles) such that we have $soc(P_i)=S_{f_i}$, when $A$ has indecomposable projective modules $P_i$ (with simple tops $S_i$) for $i=1,...,n$.
Note that in this case $A$ would be a Frobenius algebra iff $f$ is a bijection.
My example above with the $Q_f$ has this property but the $Q_f$ are in general not connected.
