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Dominic Joyce had been announcing in his article linked to at smooth algebra an application of smooth locuses and a simple version of derived smooth manifolds to moduli spaces of curves. It seems that the announced work is not out yet, but somebody kindly informed me that there are these talk notes with a little bit of further info.
I wasn’t sure where to archive tis. Ended up putting this into derived smooth manifold,
There’s a second version of Algebraic Geometry over C-infinity rings on the ArXiv now, coinciding with the appearance of a survey of the paper. A draft of a part of his book - “D-manifolds and d-orbifolds: a theory of derived differential geometry” - is available here.
Joyce has a humorous account of his research interests. And from a recent talk:
Derived manifolds were defined by David Spivak (Duke Math. J. 153, 2010), a student of Jacob Lurie. A lot of my ideas are stolen from Spivak. D-manifolds are much simpler than derived manifolds. D-manifolds are a 2-category, using Hartshorne- level algebraic geometry. Derived manifolds are an $\infty$-category, and use very advanced and scary technology – homotopy sheaves, Bousfeld localization,…
D-manifolds are a 2-category truncation of derived manifolds. I claim that this truncation remembers all the geometric information of importance to symplectic geometers, and other real people.
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