Frobenius-like Algebras, Colored Ribbon Graphs, and Extensions of TQFT

The connections between Frobenius algebras and 2D TQFTs have been widely understood for decades. The works of Dumitrescu and Mulase provide an alternate formulation of TQFT from the perspective of cell graphs. In their work, each cell graph determines a multilinear map from a tensor product of a Frobenius algebra $A$ to its underlying field $K$, and they showed that this determination only depends on the topological type of the graph and not on the graph itself. We provide an alternative proof of their classification using colored cell graphs. This new approach not only generates \textit{all} maps of a 2D TQFT (including those from tensor powers of $A$ to other tensor powers of $A$), but can further be applied to infinite-dimensional Frobenius-like algebras. We then show that this cell graph formulation is equivalent to the classic functorial approach for Frobenius-like algebras that are counital by showing that the maps that they generate are the same. After showing these two approaches are equivalent, we conjecture a formula using them in tandem to answer questions in enumerative geometry and topological recursion.