Spivak on Derived Manifolds
Posted by John Baez
Derived categories are an efficient way to “infinitely categorify” a given concept without even needing to know about $\infty$categories. They don’t do everything! But, they let you go quite far without a vast amount of effort — especially since lot of technology has been developed over the years that you can simply grab off the shelf and use.
For example, the derived category of abelian groups is not a fullfledged approach to dealing with ‘weak abelian $\infty$groups’: for those you need infinite loop spaces, or in other words, connective spectra. But, there’s a very precise sense in which derived category of abelian groups is a way of handling strict abelian $\infty$groups and weak maps between these. And that’s pretty good.
So, it’s no surprise that derived categories — and their nonabelian brethren, model categories — are taking over the mathematical universe. And once these get the foot in the door, more general infinitely categorified concepts are sure to follow. For example, we’ve recently seen the rise of derived algebraic geometry, where spectra take the place of abelian groups, and $E_\infty$ ring spectra take the place of commutative rings:
 Bertrand Toen and Gabriele Vezzosi, Algebraic geometry over model categories (a general approach to derived algebraic geometry).

Jacob Lurie, Derived algebraic geometry:
I: Stable $\infty$categories.
II: Noncommutative algebra .
III: Commutative algebra .
IV: Deformation theory.
It would be a fulltime job just keeping up with this subject.
But now we’re seeing the next phase: derived differential geometry!
I just ran into a thesis on the subject:
 David Spivak, QuasiSmooth Derived Manifolds, Ph.D. thesis, U. C. Berkeley, Spring 2007.
People interested in comparative smootheology, convenient categories of smooth spaces, stacks and categorification should take a look! Derived manifolds do a different job than general smooth spaces: they explicitly incorporate homotopical ideas. I doubt either is a substitute for the other. But it should be profitable to compare these ideas.
By the way, Spivak is a student of Peter Teichner, whose work we’ve often discussed here.
Re: Spivak on Derived Manifolds
And a nephew of the more famous mathematical writer Spivak. (Really, I think you should include David’s first name in the post title.)