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For any symmetric monoidal category $\mathcal{D}$, Lauda and Pfeiffer showedthe equivalence between the $\mathcal{D}$-valued open-closed 2-dimensionalTQFTs and the so-called knowledgeable Frobenius algebras (KFAs) in$\mathcal{D}$. Each KFA in $\mathcal{D}=\mathbf{Vec}_{\mathbb{K}}$ provides asequence of scalars indexed by the set $\mathbb{N}^2$ of diffeomorphism classesof connected endocobordisms of the empty set, given by evaluation by theassociated 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 $\textbf{1}$ satisfying$\text{End}_{\mathcal{C}_{\chi}}(\textbf{1})\cong \mathbb{K}$ -- generated by aKFA object affording this sequence. We then determine which sequences $\chi$produce semisimple abelian categories $\mathcal{C}_{\chi}$ withfinite-dimensional hom-spaces. These form a family of categories interpolatingthe categories of representations of automorphism groups of certain KFAs in$\mathbf{Vec}_{\mathbb{K}}$.

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