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!
Best Answer
Higher stacks arise very naturally before deriving, whenever we consider for example moduli problems of objects that have not just automorphisms but where the automorphisms have automorphisms. Obviously for this to make sense our objects need to have some categorical structure already! For example, if you want to make a moduli space of stacks of some kind, you naturally find a two-stack -- stacks are 2-categorical in nature, and your automorphisms may well have automorphisms. Or another way to think about this - suppose you have a group acting on a stack, and the action has nontrivial stabilizers when acting on automorphisms of points in your stack. Then the quotient is a 2-stack.. (Or you can take the quotient of a scheme by the action of a group-stack! eg for $G$ abelian take the quotient of a point by $BG$, aka $BBG$.)
[Edit: my favorite example of this (an algebraic 2-stack appearing as a quotient of a stack) is due to Ng\^o in his work on the fundamental lemma (Drinfeld coined the name "Ngo 2-stack"). You take a reductive group G, and consider the stack $G/G$. This stack carries a natural action of the group scheme of centralizers, but this is poorly behaved (centralizers jump in dimensions). Ngo realized there's a FLAT group scheme of dimension = rank of G that acts, the group scheme of regular centralizers (for each $g\in G$ take a regular element in $G$ with the same invariant polynomials, a totally elementary argument shows this actually acts on the centralizer of $g$..) Now take the quotient.. you get an algebraic 2-stack, which contains an open dense piece which is just the SCHEME $h/W$. This construction is behind Ngo's analysis of the Hitchin fibration.]
Thinking this way you might get the feeling that you'll probably never practically need more than 2-stacks. But once you start working in a homotopical context you run into higher stacks very quickly. There's a very natural example where you run into the full structure of $\infty$-stacks immediately, as explained by To\"en and Vaqui\'e: moduli of objects in a derived category. If you have a [nice] abelian category you can define a moduli stack of objects in it --- since objects have automorphisms, families of objects are naturally groupoids so you find a stack. Suppose now you have a derived category [technically you need to "enhance" it to a dg or $A_\infty$ category or something equivalent, but let's ignore this]. In this context the presence of negative self-exts of objects means that automorphisms may have automorphisms which may have automorphisms.. i.e. you find higher homotopy groups of the natural moduli functor (which lands now in simplicial sets, or topological spaces). In any case you can make precise sense of moduli of objects in a derived category, and it is an $\infty$-stack!
(Another example of the same nature is the moduli problem for derived categories themselves! here again you have higher Ext groups in the form of Hochschild cohomologies, which are explained homotopically e.g. in To\"en's Inventiones paper on derived Morita theory.)
In summary once you're in a derived context, your moduli functors naturally land not in sets, not in groupoids (which are the same as 1-truncated homotopy types) but in higher groupoids, aka homotopy types, aka simplicial sets. Such moduli problems may be representable by higher stacks.