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quasicoherent infinity-stack

Contents

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

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

or directly in terms of test spaces UU:

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

For the special case that C=(sAlg op) 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 CC an (∞,1)-site let the (∞,1)-functor

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

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

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

Definition

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

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

For instance

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

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

(…)

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

([C op,sSet] proj,loc) H:=Sh (,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 \infty-stack is a simplicial presheaf C opSSetC^{op} \to SSet on a simplicial site CC that takes values in Kan complexes and satisfies descent with respect to hypercovers in the homotopy category of CC.

Definition

The simplicial presheaf of quasicoherent \infty-stacks is

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

is given by

QC:SpecAN(AMod cof,), 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 AModA Mod on cofibrant objects and weak equivalences between them.

This appears as (ToënVezzosiII, definition 1.3.7.1.

Proposition

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

This is (ToënVezzosiII, theorem 1.3.7.2.

notice: the QCQC defined this way is not yet stabilized

References

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 \infty-stacks over perfect ∞-stacks is in

Revised on March 6, 2013 19:26:22 by Zoran Škoda (161.53.130.104)