I hope you will excuse this naive and general question. I've read from many places (e.g. Dominic Joyce's website, John Pardon's thesis, etc.) that the/a "right" foundations for many constructions in sympletic geometry, such as Fukaya category, and GromovWitten theory should be done starting from some formalism of derived differential geometry. I am aware that Dominic Joyce is still writing a series of books on this, but my question is more about let's say big picture understanding of this approach. Is there a consensus, assuming a general theory of derived manifolds, how these constructions should be done? If so, can someone explain, sacrificing the rigor needed, a high level outline of these constructions? Sorry if this is inside of say Joyce's writings.

1$\begingroup$ AIUI: for a pde with fredholm linearization, the index of linearization wants to be the tangent chain complex of the derived space of solutions. If the linearization always has zero cokernel you get a manifold. In general, a Kuranishi structure, which presents a quasismooth derived manifold. JHol curves with fixed type are derived stack. The problem is boundary, when degenerate from one type to another. Unobstructed case, a topological manifold, not a smooth manifold. General case a topological Kuranishi structure, which doesn't quite fit derived geometry. $\endgroup$– Ben WielandNov 16, 2020 at 21:11