D-manifolds, d-orbifolds and derived differential geometry: a detailed
summary

This is a long summary of the author's book "D-manifolds and d-orbifolds: a
theory of derived differential geometry", available at
http://people.maths.ox.ac.uk/~joyce/dmanifolds.html . A shorter survey paper on
the book, focussing on d-manifolds without boundary, is arXiv:1206.4207, and
readers just wanting a general overview are advised to start there.
We introduce a 2-category dMan of "d-manifolds", new geometric objects which
are 'derived' smooth manifolds, in the sense of the 'derived algebraic
geometry' of Toen and Lurie. They are a 2-category truncation of Spivak's
'derived manifolds' (see arXiv:0810.5174, arXiv:1212.1153). The category of
manifolds Man embeds in dMan as a full (2-)subcategory. We also define
2-categories dMan^b,dMan^c of "d-manifolds with boundary" and "d-manifolds with
corners", and orbifold versions of these dOrb,dOrb^b,dOrb^c, "d-orbifolds".
Much of differential geometry extends very nicely to d-manifolds and
d-orbifolds -- immersions, submersions, submanifolds, transverse fibre
products, orientations, orbifold strata, bordism, etc. Compact oriented
d-manifolds and d-orbifolds have virtual classes.
There are truncation functors to d-manifolds and d-orbifolds from essentially
every geometric structure on moduli spaces used in enumerative invariant
problems in differential geometry or complex algebraic geometry, including
Fredholm sections of Banach vector bundles over Banach manifolds, the
"Kuranishi spaces" of Fukaya, Oh, Ohta and Ono and the "polyfolds" of Hofer,
Wysocki and Zehnder in symplectic geometry, and C-schemes with perfect
obstruction theories in algebraic geometry. Thus, results in the literature
imply that many important classes of moduli spaces are d-manifolds or
d-orbifolds, including moduli spaces of J-holomorphic curves in symplectic
geometry.
D-manifolds and d-orbifolds will have applications in symplectic geometry,
and elsewhere.