Constructing new open-closed TQFTs from the interpolation of symmetric monoidal categories

For any symmetric monoidal category $\mathcal{D}$, Lauda and Pfeiffer showed the equivalence between the $\mathcal{D}$-valued open-closed 2-dimensional TQFTs and the so-called knowledgeable Frobenius algebras (KFAs) in $\mathcal{D}$. Each KFA in $\mathcal{D}={\mathbf{V}\mathbf{e}\mathbf{c}}_{\mathbb{K}}$ provides a sequence of scalars indexed by the set ${\mathbb{N}}^2$ of diffeomorphism classes of connected endocobordisms of the empty set, given by evaluation by the associated TQFT on each such cobordism class. From an arbitrary sequence $\chi =({\chi }_{g,w}{)}_{g,w\in \mathbb{N}}$, we build a symmetric monoidal category ${\mathcal{C}}_{\chi }$ – with unit object $\mathbf{\text{1}}$ satisfying ${\text{End}}_{{\mathcal{C}}_{\chi }}(\mathbf{\text{1}})\cong \mathbb{K}$ – generated by a KFA object affording this sequence. We then determine which sequences $\chi$ produce semisimple abelian categories ${\mathcal{C}}_{\chi }$ with finite-dimensional hom-spaces. These form a family of categories interpolating the categories of representations of automorphism groups of certain KFAs in ${\mathbf{V}\mathbf{e}\mathbf{c}}_{\mathbb{K}}$.