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

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

or directly in terms of test spaces $U$:

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

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

Definition

Definition

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

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

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

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

Definition

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

$$\mathrm{QC}(X):=[C^{\mathrm{op}},(\mathrm{\infty },1)\mathrm{Cat}](X,\mathrm{Mod}).$$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 $\mathrm{\infty }$-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 $𝒞$ be a monoidal model category. Write $\mathrm{CMon}(𝒞)$ for the category of commutative monoids in $𝒞$.

For instance

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

• for $𝒞=\mathrm{sAb}$, the category of abelian simplicial groups, a monoid in $𝒞$ 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 $\mathrm{CMon}(𝒞)$.

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

(…)

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

$$([C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}{)}^{\circ }\simeq H:={\mathrm{Sh}}_{(\mathrm{\infty },1)}(C).$$([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 $\mathrm{\infty }$-stack is a simplicial presheaf $C^{\mathrm{op}}\to \mathrm{SSet}$ 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 $\mathrm{\infty }$-stacks is

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

is given by

$$\mathrm{QC}:\mathrm{Spec}A\mapsto N(A{\mathrm{Mod}}_{\mathrm{cof},\sim }),$$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 $A\mathrm{Mod}$ on cofibrant objects and weak equivalences between them.

This appears as (ToënVezzosiII, definition 1.3.7.1.

Proposition

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

This is (ToënVezzosiII, theorem 1.3.7.2.

notice: the $\mathrm{QC}$ defined this way is not yet stabilized

Related concepts

• quasicoherent sheaf

• quasicoherent $\mathrm{\infty }$-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 $\mathrm{\infty }$-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