A derived stack $X$ is an ∞-stack – an (∞,1)-sheaf – on an (∞,1)-site $C$. For example, $C$ is most often the (∞,1)-category of simplicial commutative rings or E-∞ ring spectra (with the Zariski or etale topology)).
One says derived stack in order to distinguish from the more restrictive notion of an ∞-stack on a 1-categorical site, such as for instance described at topological ∞-groupoid.
For recall that a sheaf is a functor $F : C^{op} \to Set$ satisfying some descent-condition. So there are two steps in which the notion of sheaf may be categorified:
The categorification of the codomain leads to the notion of stacks when Set replaced by Grpd, and further to ∞-stacks, when sets are replaced by ∞-groupoids.
But there is no natural reason why the domain should in general remain a 1-category if one passes to an ∞-categorical-context. A derived stack is a generalization of the notion of sheaf where both domain and codomain are taken to be $\infty$-categorical.
Following the general logic of models for ∞-stack (∞,1)-toposes, derived stacks are typically modeled by the model structure on SSet-enriched presheaves on an SSet-site or model site $C$. In such a model a derived stack is represented by an SSet-enriched functor $F : C^{op} \to SSet$ from an SSet-enriched category $C$ to SSet that satisfies a descent condition.
One general idea for the use of higher and derived stacks is that
passing to a higher categorical codomain – i.e. from Set-values sheaves to higher groupoid valued sheaves – is a means to obtain good colimits, colimits that do not lose information. For instance
in the category Diff of manifolds the quotient by a non-free action of a group may not exist
in sheaves in $[Diff^{op},Set]$ it will exist, but will have the wrong properties in general with respect to some operations such as taking cohommology,
while finally in stacks $[Diff^{op}, Grpd]$ it exists as the corresponding smooth action groupoid or orbifold and in this form rembers in terms of the isomorphisms how the quotient was obtained. The cohomology of the stack is then indeed the equivariant cohomology of the original manifold.
similarly passing to higher categorical domain – i.e. from presheaves on categories to presheaves on higher categories, is analogously a means to ensure that good limits exist.
A detailed illustration and motivation of the need of these “good limits” that don’t forget the way they were formed is
in the introductory section of
in the introductory section of
A derived refinement of the ordinary site CRing${}^{op}$ of formal duals to commutative tings is the SSet-site $SCRing^{op}$ of simplicial rings. Derived stacks on this site are studied in derived algebraic geometry.
An introduction to this is for instance in chapter 5 of the lecture notes
There are various slight variations of this. For instance using the monoidal Dold-Kan correspondence, simplicial rings may be replaced with non-positively graded cochain dg-algebras.
One application of derived stacks on $(dgAlg^-)^{op}$ is the BV-BRST formalism in physics.
A derived refinement of the ordinary site $\mathbb{L}$ of smooth loci is the SSet-site $cs \mathbb{L}$ of cosimplicial smooth loci. Derived stacks on this are the objects in a theory that could be called derived synthetic differential geometry?.
One obvious but notable phenomenon that occurs in derived stacks in general, but not in ∞-stacks over a 1-categorical site, is that under the Yoneda embedding for (∞,1)-categories a categorically discrete object, i,e. a 0-truncated object may be mapped to a higher categorical object.
Consider specifically an ordinary site $C$ and let $csC = [\Delta,C]$ be the corresponding SSet-site of cosimplicial objects. Write
for the composite that first regards objects of $C$ as constant cosimplicial objects and then applies the derived functor of the enriched Yoneda embedding, i.e. the functor
for $Q$ a fibrant and $P$ a cofibrant replacement functor. Then of course the simplicial presheaf $\mathbf{Y}(U)$ for $U \in C$ may in general take values in simplicial sets with nontrivial higher simplicial homotopy groups, to the extent that the SSet-hom-objects $\mathbf{Y}(U) : V \mapsto [\Delta,C](V,U)$ in $[\Delta,C]$ are nontrivial.
This cannot happen for ∞-stacks over 1-categorical site. For instance a topological space regarded as a topological ∞-groupoid is always an 0-truncated object as an ∞-stack.
A notable example of this is the case where $C =$ CRing and $U = Spec R$. While ordinarily this is 0-categorical, when regarded as a derived stack on formal duals of simplicial rings this has in general a nontrivial free loop space object, i.e. a nontrivial homotopy pullback of the form
where, as indicated, the free loop space object is something like the de Rham space of $Spec R$. This means that regarded as a derived stack, the space $Spec R$ becomes an ∞-groupoid whose morphisms are given by infinitesimal paths in the orighinal space.
By the $\infty$-erspective on Hochschild cohomology (as discussed there) this implies a bunch of nice relations. Details are in
The fact is also mentioned and used in passing every now and then (e.g. p. 9) in
But there must be a better reference, somewhere.
An overview is provided in
A set of lecture notes on the model structure on simplicial presheaves with an eye towrads algebraic sites and derived algebraic geometry is
Details modeled on simplicial categories have been developed in the series of articles by Toën and Vezossi.
This article generalizes the notion of site and model structure on simplicial presheaves from 1-categorical sites to simplicial sites and hence to a model structure on SSet-enriched presheaves:
Of central interest in derived algebraic geometry is the simplicial site of simplicial algebras, which generalizes the familiar site of algebra used in algebraic geometry. This is introduced and studied in
Further developments in this direction are in
The unifying picture, in particular independent of the choice of model for the (infinity,1)-categories is presented in
Derived ($\infty$-)stacks are currently mostly, maybe exclusively, studied on algebraic sites $S$, where the category $S^{op} :=$ Alg is replaced with a category of “$\infty$-algebras” of sorts. The theory of these $\infty$-algebras is described in great detail in
Concretely the need for the site of simplicial ring objects is discussed in the introduction of
and in the introduction of
The proof that simplicial algebras are Quillen equivalent of differential graded algebras – so that derived stacks on simplicial algebras are the same as derived stacks on DGAs – is in