In Toen's and Vezzosi's article From HAG to DAG: derived moduli stacks a kind of definition of DAG is given. I am not an expert and can't see what's the relation between DAG and the motivic cohomology ideas of Voevodsky (particularly his category $DG$). It would be nice if somebody could explain that to me.
I addition I am very keen on seeing how these ideas can be used in an explicit example. If I should explain someone why motivic cohomology is a good thing, I would certainly mention the proof of the Milnor conjecture. But can one see the use of derived ideas in a more explicit and down-to-earth example?
Best Answer
There is a very general nice pattern here:
Let $C$ be a category of test spaces on which we want to model more general spaces. Then
Morel-Voevodsky take C to be the Nisnevich site. Then $Sh_{(\infty,1)}(C)$ is the the ∞-topos whose intrinsic cohomology is motivic cohomology.
Here C happens to be just an ordinary category. More generally, we could take C to be an ∞-category itself. In that case $Sh_{(\infty,1)}(C)$ could be called the ∞-topos of derived stacks. What Toen Vezzosi do in HAG I and II is to provide a model-cateory theoretic presentation of this. That's why the articles look like they are hard to read: this is a component based way to describe an abstract elegant concept.
Now, the objects in $Sh_{(\infty,1)}(C)$ are "very general" spaces modeled on $C$. There is a chain of ∞-subcategories of more "tame" spaces inside, though:
first there are those ∞-stacks on C which are represented by an ∞-stack with a C-valued structure sheaf. This are the structured ∞-topos, that generalize the notion of ringed spaces.
and then among these are those that are locally equivalent to objects in C. These are the generalized schemes or "derived scheme" if C is suitably ∞-categorical.
The pattern here is reviewed at notions of space.
In principle one could consider such "derived schemes" also in the Morel-Voevodsky gros ∞-topos of ∞-stacks on the Nisnevich site, it just seems that so far nobody looked into this. But there are all kinds of examples of test space categories C that people still have to push through this general nonsense. For instance we can take C to be simply the category of smooth manifolds. Then the above generalities spit out the notion of a derived smooth manifold.
The punchline being: all these things unify in one grand picture. It is not Toen/Vezzosi/Lurie derived geometry on one hand and Morel/Voevodsky cohomology on the other. Instead all this is parts of one big picture.
And also, if I may say this here, this big picture really is hardly restricted to algebraic geometry. Lurie's notion of space is much, much more general. It describes GEOMETRY. Of whatever sort.