The notion of quasicoherent ∞-stack of modules is the anlog in (∞,1)-topos theory of the notion of quasicoherent sheaf in topos theory.
A quasicoherent sheaf on a scheme is is equivalently a morphism of stacks into the canonical stack Mod of modules, which corresponds to the bifibration over the category of commutative rings/algebras: this is the tangent category of CRing.
This general abstract description of quasi-coherent sheaves has a fairly direct generalization to (∞,1)-topos theory over arbitrary (∞,1)-sites:
for any (∞,1)-site, the tangent (∞,1)-category is the bifibration whose fibers over an object plays the role of the ∞-groupoid of modules over . Under the (∞,1)-Grothendieck construction this corresponds to an (∞,1)-presheaf
or directly in terms of test spaces :
For the special case that opposite (∞,1)-category presented by the model structure on simplicial algebras over a characteristic 0 ground field we have (with example 8.6 of Stable ∞-Categories ) this reproduces the notion of quasicoherent -stack as considered in dg-geometry.
Let be the (∞,1)-category of (∞,1)-sheaves over . Notice that this sits inside .
For , the -category of quasi-coherent -stacks on is the (∞,1)-category of (∞,1)-functors
Model category theoretic presentation
For derived geometry modeled on formal duals of algebras over an operad, the model category presentation of quasi-coherent -stacks is locally given by a model structure on modules over an algebra over an operad.
The following considers the special case of the commutative operad.
For commutative monoids
Let be a monoidal model category. Write for the category of commutative monoids in .
In (ToënVezzosi, I, ToënVezzosi, II) are discussed structures of a model site/sSet-site on .
For every commutative monoid There is naturally a model category structure on the category of -modules in .
Let the local model structure on sSet-presheaves that presents the (∞,1)-category of (∞,1)-sheaves/∞-stacks on
as described as models for ∞-stack (∞,1)-toposes.
So in this model an -stack is a simplicial presheaf on a simplicial site that takes values in Kan complexes and satisfies descent with respect to hypercovers in the homotopy category of .
The simplicial presheaf of quasicoherent -stacks is
is given by
where on the right we have the nerve of the non-full subcategory of the model category on cofibrant objects and weak equivalences between them.
This appears as (ToënVezzosiII, definition 220.127.116.11.
This is indeed an ∞-stack on the model site
This is (ToënVezzosiII, theorem 18.104.22.168.
notice: the defined this way is not yet stabilized
The general discussion of the tangent (∞,1)-category is in
The model category theoretic presentation over model sites of commutative monoids is discussed in
A discussion specific to dg-geometry with an emphasis on the geometric ∞-function theory of quasicoherent -stacks over perfect ∞-stacks is in