TITLE: Complicating an Example by Toen (motivations for DAG)
QUESTION [14 upvotes]: I'm trying to read (the introduction of) a survey by Toen on Derived Algebraic Geometry, specifically the "Simplicial Presheaves and Derived Algebraic Geometry" one.
He motivates the introduction of DAG as a means to construct moduli spaces. His example is the moduli of linear representations of a group admitting a finite presentation.
Now, DAG (AFAIU) enlarges the theory of stacks in two directions: a derived bit and a stacky bit. The derived bit concerns replacing rings with more general ring-like objects. The stacky bit comes from using stacks of oo-categories.
His motivation for the `derived' direction comes from the fact that in constructing a moduli space he gets a tensor product, which isn't considering higher Tor's. (I guess I'm sort of OK with that, but for no concrete reason)
His motivation for the `stacky' direction comes from taking quotients of a 'rigidified' moduli problem, which admits a moduli space (the aforementioned tensor product). The problem here is the usual fact that you want to remember the isomorphism groups of objects.
Unfortunately this only motivates us to introduce stacks, not oo-stacks.

So my question is: Can we complicate this example a bit more (but hopefully not too much) so that we need to use higher stacks?

I understand that if we have already enhanced our geometry along the derived direction then we actually need oo-stacks to keep track of notions of equivalence which are weaker than isomorphisms.
So a parallel question might be:

if we don't derive our geometry first, do we need to introduce higher stacks?

Comments on the derived are also very much appreciated, thanks!

REPLY [2 votes]: The moduli of linear categories (or abelian categories) in naturally a $2$-stack. You can take a look at the PhD thesis (in french) of Mathieu Anel (a former student of Bertrand Toën). 
When you compute the tangent complex of this $2$-stack then you get the $2$-truncation of the Hochschild complex. If you consider the corresponding derived stack then you get the full Hochschild complex. 
By the way, higher stacks are introduced to allow quotients, while derived schemes are introduced to allow fiber products.