Interpolation of open-closed TQFTs

For any symmetric monoidal category $C$, Lauda and Pfeiffer showed the equivalence between the $C$-valued open-closed 2-dimensional TQFTs and the so-called knowledgeable Frobenius algebras (kFAs) in $C$. A kFA in the category of finite-dimensional vector spaces over a field $K$ provides a sequence of scalars indexed by the set $N^2$ of diffeomorphism classes of connected endocobordisms of the empty set, given by evaluation of the associated TQFT on each such cobordism class. More generally, from an arbitrary sequence $\chi ={\left({\chi }_{g,w}\right)}_{g,w\in N}$, we show how to build a symmetric monoidal category $〈T_{\text{kFA}}|\chi 〉$, with unit object 1 satisfying $\text{End}\left(\text{1}\right)\cong K$, generated by a kFA affording this sequence. We then determine which sequences χ produce semisimple abelian categories $〈T_{\text{kFA}}|\chi 〉$ with finite-dimensional hom-spaces. These categories generalise results of Deligne concerning the interpolation of families of categories of representations such as $\text{Rep}\left(S_d\right)$, $\text{Rep}\left(O_d\right)$, and $\text{Rep}(S_{\eta }\wr PG{L}_d)$.