Dioperads, Frobenius monoidal functors and integration along fibers

Dioperads encode algebraic structures with several input and output, generalizing operads. In the same way lax monoidal functors are exactly those preserving algebras over operads, I will explain that Frobenius monoidal functors are exactly those preserving algebras over dioperads.
In a second part, I shall describe how to construct (shifted) Frobenius monoidal structures given a certain orientation data, analogous to the procedure of integration along fibers induced by Poincaré duality. This construction arose from a question in derived Poisson geometry, which requires an ∞-categorical generalizing of the previous result. Time depending, I will discuss this motivation and the homotopical difficulties that stand in the way.
This is joint work with Valerio Melani.