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?



What is DAG and what has it to do with the ideas of Voevodsky?
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 


* a "very general space" modeled on $C$ is an object in the [gros](http://ncatlab.org/nlab/show/petit+topos) [∞-topos](http://ncatlab.org/nlab/show/(infinity%2C1)-topos) $Sh\_{(\infty,1)}(C)$ of ∞-stacks on $C$.


Morel-Voevodsky take C to be the [Nisnevich site](http://ncatlab.org/nlab/show/Nisnevich+site). Then $Sh\_{(\infty,1)}(C)$ is the the ∞-topos whose [intrinsic cohomology](http://ncatlab.org/nlab/show/cohomology) is [motivic cohomology](http://ncatlab.org/nlab/show/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 stack](http://ncatlab.org/nlab/show/derived+stack)s. 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](http://ncatlab.org/nlab/show/structured+(infinity%2C1)-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](http://ncatlab.org/nlab/show/generalized+scheme#definition_in_the_variant_of_lurie_5) or "derived scheme" if C is suitably ∞-categorical.


The pattern here is reviewed at [notions of space](http://ncatlab.org/nlab/show/space#NotionsOfSpace).


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](http://ncatlab.org/nlab/show/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](http://ncatlab.org/nlab/show/space#NotionsOfSpace) is much, much more general. It describes GEOMETRY. Of whatever sort.


