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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 Gromov-Witten 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.

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    $\begingroup$ 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 quasi-smooth 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 Wieland
    Commented Nov 16, 2020 at 21:11

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I would recommend that you read the introduction of Pelle Steffens's foundational preprint on Derived $C^\infty$ geometry. The preprint is 205 pages long, but the introduction is "only" 35 pages :-)

In a subsequent preprint on Representability of elliptic moduli problems, he actually implement the strategy sketched in the previous one. The introduction of this preprint is shorter (less than 10 pages) than the introduction to the previous one, and goes more straight to the point, still sketching the general strategy rather well (it can be better suited especially if you're already convinced that $\infty$-categories and derived differential geometry can be useful in this context).

There is also an excellent survey of John Pardon where he explains similar ideas, maybe from a different angle. It has the huge advantage of being rather short (20 pages).

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  • $\begingroup$ Is there anything in differential geometry similar to DG-categories in derived algebraic geometry? $\endgroup$
    – Z. M
    Commented Dec 14 at 20:02

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