nLab quasicoherent infinity-stack

Idea

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 $X$ is is equivalently a morphism of stacks $X \mapsto Mod$ into the canonical stack Mod $: Spec A \mapsto A Mod$ of modules, which corresponds to the bifibration $Mod \to CRing$ 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 $C$ any (∞,1)-site, the tangent (∞,1)-category $T (C^{op}) \to C^{op}$ is the bifibration whose fibers over an object $A \in C^{op}$ plays the role of the ∞-groupoid of modules over $A$. Under the (∞,1)-Grothendieck construction this corresponds to an (∞,1)-presheaf $\infty Mod : C^{op} \to \infty Cat$

$$\infty Mod : Spec R \mapsto Stab( C^{op}/R )$$

or directly in terms of test spaces $U$:

$$\infty Mod : U \mapsto Stab( U/C ) \,.$$

For the special case that $C = (sAlg^{op})^\circ$ 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 $\infty$-stack as considered in dg-geometry.

Definition

Definition

For $C$ an (∞,1)-site let the (∞,1)-functor

$$Mod_C : C^{op} \to (\infty,1)Cat$$

be that corresponding under the (∞,1)-Grothendieck construction to the tangent (∞,1)-category $T C^{op} \to C^{op}$.

Let $\mathbf{H} := Sh_{(\infty,1)}(C)$ be the (∞,1)-category of (∞,1)-sheaves over $C$. Notice that this sits inside $[C^{op}, (\infty,1)Cat]$.

Definition

For $X \in \mathbf{H}$, the $(\infty,1)$-category of quasi-coherent $\infty$-stacks on $X$ is the (∞,1)-category of (∞,1)-functors

$$QC(X) := [C^{op}, (\infty,1)Cat](X, Mod) \,.$$

Model category theoretic presentation

For derived geometry modeled on formal duals of algebras over an operad, the model category presentation of quasi-coherent $\infty$-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 $\mathcal{C}$ be a monoidal model category. Write $CMon(\mathcal{C})$ for the category of commutative monoids in $\mathcal{C}$.

For instance

• for $\mathcal{C} =$ Ab a monoid object in $\mathcal{C}$ is an ordinary ring and its formal dual an ordinary affine scheme;

• for $\mathcal{C} = sAb$, the category of abelian simplicial groups, a monoid in $\mathcal{C}$ is a simplicial ring and its formal dual is one notion of affine derived scheme.

In (ToënVezzosi, I, ToënVezzosi, II) are discussed structures of a model site/sSet-site on $CMon(\mathcal{C})$.

For every commutative monoid $A \in \mathcal{C}$ There is naturally a model category structure on the category $A Mod$ of $A$-modules in $\mathcal{C}$.

(…)

Let $[C^{op},sSet]_{proj,loc}$ the local model structure on sSet-presheaves that presents the (∞,1)-category of (∞,1)-sheaves/∞-stacks on $C$

$$([C^{op}, sSet]_{proj,loc})^\circ \simeq \mathbf{H} := Sh_{(\infty,1)}(C) \,.$$

as described as models for ∞-stack (∞,1)-toposes.

So in this model an $\infty$-stack is a simplicial presheaf $C^{op} \to SSet$ on a simplicial site $C$ that takes values in Kan complexes and satisfies descent with respect to hypercovers in the homotopy category of $C$.

Definition

The simplicial presheaf of quasicoherent $\infty$-stacks is

$$QC : C^{op} \to sSet$$

is given by

$$QC : Spec A \mapsto N( A Mod_{cof, \sim} ) \,,$$

where on the right we have the nerve of the non-full subcategory of the model category $A Mod$ on cofibrant objects and weak equivalences between them.

This appears as (ToënVezzosiII, definition 1.3.7.1.

Proposition

This $QC$ is indeed an ∞-stack on the model site $C := Comm(\mathcal{C})^{op}$

This is (ToënVezzosiII, theorem 1.3.7.2.

notice: the $QC$ defined this way is not yet stabilized

Related concepts

• quasicoherent sheaf

• quasicoherent $\infty$-stack

References

The general discussion of the tangent (∞,1)-category is in

• Jacob Lurie, Deformation Theory .

The model category theoretic presentation over model sites of commutative monoids is discussed in

• Bertrand Toën, Gabriele Vezzosi, Homotopical Algebraic Geometry I: Topos theory (arXiv:0207028)

• Bertrand Toën, Gabriele Vezzosi, Homotopical Algebraic Geometry II: geometric stacks and applications (arXiv:0404373)

A discussion specific to dg-geometry with an emphasis on the geometric ∞-function theory of quasicoherent $\infty$-stacks over perfect ∞-stacks is in

• Ben-Zvi, John Francis, David Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry (arXiv)

category: algebraic geometry