(Discussion revised December 2015.) I am writing a new book, which is under contract to be published by Oxford University Press, hopefully to appear by the end of 2016. If you are interested, you could start with the following survey paper:D. Joyce, 'An introduction to d-manifolds and derived differential geometry', pages 230-281 in L. Brambila-Paz, O. Garcia-Prada, P. Newstead and R.P. Thomas, editors, 'Moduli spaces', London Mathematical Society Lecture Note Series 411, Cambridge University Press, 2014.
Here too is a "detailed summary" of the book, a longer survey paper, which basically corresponds to Chapter 1 and Appendix A of the preliminary version of the book, and contains more-or-less all the main definitions and theorems, but no proofs.
D. Joyce, 'D-manifolds, d-orbifolds and derived differential geometry: a detailed summary', arXiv:1208.4948, version 2, December 2012. 173 pages.
You can download a preliminary version of the book (from 2012) here:
'D-manifolds and d-orbifolds: a theory of derived differential geometry', pdf file. 768 pages.Warning: the final OUP version of the book will use a different definition of d-manifolds to the three 2012 papers/books above, which is more-or-less equivalent, but shorter and nicer to use; and it will have stronger theorems, will be better than the preliminary version in lots of other ways, and may even be shorter. I am not going to make the OUP version of the book available online, or any draft more recent than 2012, as this would make OUP cross; so if you want to read it when it is published, you had better ask your librarian to order a copy, or even better buy one yourself.
You can download the slides of a talk I gave on it in London in December 2015 here:
'Derived differential geometry'. pdf file
In August 2015 I gave a 14 hour lecture course on 'Derived
differential geometry' to a summer school in Aarhus. You can download
the slides of the lectures from the course
Here is a survey paper on C∞-schemes and C∞-algebraic geometry which may be useful: arXiv:1104.4951.
I have also written a long paper arXiv:1409.6908 giving a new definition of Kuranishi space, which is summarized in the survey paper arXiv:1510.07444. This defines an ordinary category MKur of 'M-Kuranishi spaces' which is equivalent to the homotopy category Ho(dMan) of the strict 2-category dMan of d-manifolds. It defines a weak 2-category Kur of 'Kuranishi spaces', which is equivalent as a weak 2-category to the strict 2-category dOrb of d-orbifolds. It also defines categories MKurb, MKurc of 'M-Kuranishi spaces with boundary' and 'with corners', and weak 2-categories Kurb, Kurc of 'Kuranishi spaces with boundary' and 'with corners'. These are not equivalent to Ho(dManb), Ho(dManc), dOrbb, dOrbc for the definitions of dManb, dManc, dOrbb, dOrbc in the 2012 preliminary version of the d-manifolds book above, but they will be equivalent using the definitions in the final OUP version.Thus, Kuranishi spaces, with my new definition, are essentially equivalent to d-manifolds and d-orbifolds, and can be used instead for many purposes. Kuranishi spaces are more elementary, but d-manifolds and d-orbifolds have some technical advantages -- for instance, being embedded in larger 2-categories of 'd-spaces' and 'd-stacks' which are closed under fibre products.
Here is a brief description of what the d-manifolds book is about. See the reference list at the end for relevant literature.
"Kuranishi spaces" are a class of geometric spaces introduced in 1990 by Fukaya and Ono, as the geometric structure on moduli spaces of J-holomorphic curves in a symplectic manifold, and used in the work of Fukaya, Oh, Ohta and Ono on Lagrangian Floer cohomology and Fukaya categories. Although their definition was sufficient for their applications, it did not give a very satisfactory notion of geometric space -- notions of morphisms, or even of when two Kuranishi spaces are "the same", are not well behaved -- so the theory of Kuranishi spaces was never developed very far. The book began as a project to find the "right" definition of Kuranishi space, which I believe I have done.
The book describes and studies a new class of geometric objects I call "d-manifolds". D-manifolds are a kind of "derived" smooth manifold, where "derived" is in the sense of the derived algebraic geometry of Jacob Lurie, Bertrand Toen, etc. The closest thing to them in the literature is the "derived manifolds" of David Spivak. But d-manifolds are rather simpler than Spivak's derived manifolds -- d-manifolds form a 2-category which is constructed using fairly basic techniques from algebraic geometry, but derived manifolds form an ∞-category (simplicial category) which uses advanced ideas like homotopy sheaves and Bousfeld localization.
Following Spivak, my d-manifolds are based on the idea of "C∞-schemes". These were invented in Synthetic Differential Geometry (see Dubuc, 1981, and my 2010 paper below), and are a way of applying the methods of Algebraic Geometry in the world of real smooth geometry to get a large class of geometric objects which include smooth manifolds, but also many singular spaces.
Manifolds are examples of d-manifolds -- that is, the category of manifolds embeds as a subcategory of the 2-category of d-manifolds -- but d-manifolds also include many spaces one would regard classically as singular or obstructed. A d-manifold has a virtual dimension, an integer, which may be negative. Almost all the main ideas of differential geometry have analogues for d-manifolds -- submersions, immersions, embeddings, submanifolds, orientations, transverse fibre products, and so on -- but the derived versions are often stronger. For example, the intersection of two submanifolds in a manifold exists as a manifold if the intersection is transverse, but it always exists as a d-manifold. There are also good notions of d-manifolds with boundary and d-manifolds with corners, and orbifold versions of all this, d-orbifolds. I claim that the (morally and aesthetically) "right" definition of Kuranishi space in the work of Fukaya-Oh-Ohta-Ono is that they are d-orbifolds with corners.
A useful property of d-manifolds and d-orbifolds is that they have well-behaved virtual cycles or virtual chains. So, for example, if X is a compact oriented d-manifold of virtual dimension k, and Y is a manifold, and f : X →Y is a 1-morphism, then we can define a virtual class [X] in the homology group Hk(Y; Z), which is unchanged under deformations of X,f. This will be important in applications of d-manifolds and d-orbifolds.
Many important areas of mathematics involve "counting" moduli spaces of geometric objects to define enumerative invariants or homology theories -- for instance, Donaldson and Seiberg-Witten invariants for 4-manifolds, Donaldson-Thomas invariants of Calabi-Yau 3-folds, Gromov-Witten invariants in algebraic or symplectic geometry, instanton Floer cohomology, Lagrangian Floer cohomology, contact homology, Symplectic Field Theory, Fukaya categories, ... In all of these (at least over C for the algebraic geometry cases, and away from reducible connections, etc.) the moduli spaces concerned will be oriented d-manifolds or d-orbifolds, and the counting can be done using virtual cycles or chains. So, d-manifolds and d-orbifolds provide a unified way of looking at these counting problems. There are truncation functors from geometric structures currently used to define virtual classes to d-manifolds and d-orbifolds. For instance, any moduli space of solutions of a smooth nonlinear elliptic p.d.e. on a compact manifold is a d-manifold. In algebraic geometry, a C-scheme with a perfect obstruction theory can be made into a d-manifold. In symplectic geometry, Kuranishi spaces and polyfold structures on moduli spaces of J-holomorphic curves induce d-orbifold structures.
D. Borisov, 'Derived manifolds and Kuranishi models', arXiv:1212.1153.D. Borisov and J. Noel, 'Simplicial approach to derived differential manifolds', arXiv:1112.0033.
K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, 'Lagrangian intersection Floer theory - anomaly and obstruction', Parts I and II. AMS/IP Studies in Advanced Mathematics, 46.1 and 46.2, A.M.S./International Press, 2009.
K. Fukaya and K. Ono, 'Arnold Conjecture and Gromov-Witten invariant', Topology 38 (1999), 933-1048.
D. Joyce, 'Algebraic Geometry over C∞-rings', arXiv:1001.0023, 2010.
J. Lurie, 'Derived Algebraic Geometry I: Stable ∞-categories', math.CT/0608228, 2006.
J. Lurie, 'Derived Algebraic Geometry V: Structured spaces', arXiv:0905.0459, 2009.
D.I. Spivak, 'Derived smooth manifolds', Duke Mathematical Journal 153 (2010), 55-128. arXiv:0810.5174.