Despite not working in algebra I’m a sucker for it. So when I came across an article on derived smooth manifolds I knew I had to have a look.
The phrase “derived X”, when used in a mathematical context usually refers to some category of objects which is built out of a category describing the thing X. The new category will have been constructed in such a way that some nice property that the X’s don’t have is now true in the new category.
The most common example are derived categories and derived functors. The motivation here is that certain operations, like the tensor product, which doesn’t preserve short exact sequences. In this case one can form the derived tensor functor. This function is referred to as tor. It preserves short exact sequences of the objects of categories that it is defined from. This is an important point since tor operates on long exact sequences of objects.
The transition from the tensor product to tor involves replacing an object by some long exact sequence which some how represent the original object. The idea here is that the information about the object that we really want is entirely contained in the long exact sequence. We can then translate the action of the tensor product to the action of the tensor product on the exact sequence. By making suitable choices in the construction of the sequence we can ensure that the tensor product works “correctly” on the sequence. Once some technical details are taken care of this idea can be applied to an entire category.
In any case the principle here is that “derived” kind of means “fix”. This is an interesting concept in mathematics as there are usually very good intuitive reasons for the definitions we have.
Spivak takes these ideas and applies them to a type of K-theory defined using cobordisms. The details aren’t important. What is important is that a operation on elements in the K-theory doesn’t work properly. In this case it is the cup product. The cup product is related to the intersection of manifolds and this usually requires some form of appeal to transversality of the manifolds. This is a problem is the cup product is very useful.
Spivak takes a similar approach to the usual derived approach by “dividing out” by homotopies. To my untrained eye I guessing that he shows how transversality can by altered homotopically and therefore allows for the cup product to be defined. Of course he does slightly more than this as his derived category is larger than the category of smooth manifolds. In this way his construction suggests a new type of object to replace manifolds. Predictably he uses rings to define objects that behave similarly to affine schemes. His derived manifolds are defined in a similar way to affine schemes and have strong relationships to locally ringed spaces.
I find this extremely interesting. There are hugely important relationships between manifolds and algebras over them. Moreover giving an algebraic definition of a more generalised manifold structure might allow for some of the impressive techniques of algebra to be applied.
So I guess in short I like the method and idea of the paper but am not so much interested in the specific results since I, currently, don’t care so much about coobordism. The paper has a number of strong connections with other powerful results and a growing body of work on ringed spaces. The result being that this kind of construction may prove to be very valuable.