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In my Montreal lecture notes of 1988, it was suggested that the theory of linear quantum groups can be presented in the framework of the category of  quadratic algebras  (imagined as algebras of functions on ‘quantum linear spaces), and quadratic algebras of their inner (co)homomorphisms.  Soon it was understood (Getzler and Jones, Ginzburg, Kapranov, Kontsevich, Markl, Vallette    e  ta  l   .) that the class of  quadratic operads  can be introduced and the main theorems about quadratic algebras can be generalised to the level of such operads, if their components are  linear spaces  (or objects of more general monoidal categories.)  When quantum cohomology entered the scene, it turned out that the basic tree level (genus zero) (co)operad of quantum cohomology not only is  quadratic  one, but its  components are themselves quadratic algebras .  In this short note, I am studying the interaction of quadratic algebras structure with operadic structure in the context of enriched category formalism due to Kelly  et al .

Higher structures, quantum groups and genus zero modular operad