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In a previous work by the author it was shown that every finite dimensionalalgebraic structure over an algebraically closed field of characteristic zero Kgives rise to a character $K[X]_{aug}\to K$, where $K[X]_aug$ is a commutativeHopf algebra that encodes scalar invariants of structures. This enabled us tothink of some characters $K[X]_{aug}\to K$ as algebraic structures with closedorbit. In this paper we study structures in general symmetric monoidalcategories, and not only in $Vec_K$. We show that every character $\chi :K[X]_{aug}\to K$ arises from such a structure, by constructing a category$C_{\chi}$ that is analogous to the universal construction from TQFT. We thengive necessary and sufficient conditions for a given character to arise from astructure in an abelian category with finite dimensional hom-spaces. We callsuch characters good characters. We show that if $\chi$ is good then $C_{\chi}$is abelian and semisimple, and that the set of good characters forms aK-algebra. This gives us a way to interpolate algebraic structures, and alsosymmetric monoidal categories, in a way that generalizes Deligne's categories$Rep(S_t)$, $Rep(GL_t(K))$, $Rep(O_t)$, and also some of the symmetric monoidalcategories introduced by Knop. We also explain how one can recover the recentconstruction of 2 dimensional TQFT of Khovanov, Ostrik, and Kononov, by themethods presented here. We give new examples, of interpolations of thecategories $Rep(Aut_{O}(M))$ where $O$ is a discrete valuation ring with afinite residue field, and M is a finite module over it. We also generalize theconstruction of wreath products with $S_t$, which was introduced by Knop.

Interpolations of monoidal categories and algebraic structures by invariant theory