Schreiber
∞-vector bundle

This entry is about the general notion of quasicoherent ∞-stacks, the categorification of the notion of quasicoherent sheaf.

While not every ordinary quasicoherent sheaf is equivalent to a vector bundle, every quasicoherent ∞-stack is weakly equivalent to a complex of projective modules. This may be thought of as an $\mathrm{\infty }$-vector bundle.

Abstract

We consider a natural generalization of the notion of vector space/module, and vector bundle/quasicoherent sheaf of modules from ordinary category theory to the higher category theory of (∞,1)-categories. The notion is based based on

• Lurie’s Deformation Theory

and on

• Ben-Zvi/Francis/Nadler’s geometric ∞-function theory.

This yields a general notion of quasicoherent ∞-stacks of modules – ∞-vector bundles – in any ∞-stack (∞,1)-topos.

The general definition subsumes as a special case the “derived” quasicoherent sheaves considered by Ben-Zvi/Francis/Nadler. More generally, the definition yields a notion of smooth $\mathrm{\infty }$-vector bundles on spaces in any smooth (∞,1)-topos. There is an evident notion of $\mathrm{\infty }$-vector bundles associated to any principal ∞-bundle and from the theory of differential nonabelian cohomology we obtain a notion of smooth $\mathrm{\infty }$-vector bundles with connection.

Idea

The classical notion of vector bundle generalizes to the classical notion of quasicoherent sheaf, which is more well behaved under a number of natural operations.

But in fact, a closer look reveals that the notion of quasicoherent sheaf is of a fundamental simplicity that is not well reflected in its name: a classical quasicoherent sheaf of modules on a scheme $X$ is nothing but a morphism of stacks $X\mapsto \mathrm{Mod}$ into the canonical stack $\mathrm{Mod}:\mathrm{Spec}A\to A\mathrm{Mod}$ of modules. (For details on this see the discussion at quasicoherent sheaf.)

Of this there is an obvious categorification: for $\mathrm{\infty }\mathrm{Mod}$ any (∞,1)-category valued ∞-stack on any site $C$, and for $X$ any ordinary (i.e. ∞Grpd-valued) ∞-stack we may think of a morphism $X\to \mathrm{\infty }\mathrm{Mod}$ as defining a “quasicoherent” $\mathrm{\infty }$-module on $X$, with respect to the notion of module encoded by $\mathrm{\infty }\mathrm{Mod}$.

But moreover, after passage to (∞,1)-categories, a little miracle happens: there is now a canonical notion of $\mathrm{\infty }\mathrm{Mod}$ for any given site $C$: if we regard the objects of $C$ as test spaces and hence the objects of the oppposite (∞,1)-category $C^{\mathrm{op}}$ as function rings on test spaces, then the $(\mathrm{\infty }\mathrm{,1})$-category $\mathrm{\infty }\mathrm{Mod}(R)$ of modules over the function ring $R$ may be identitfied simply with the stabilization of the overcategory of $C$ over $R$:

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

The Cartesian fibration classified by this is the tangent (∞,1)-category $T_C\to C$ of the site. This, and its relations to modules, is discussed in Lurie’s Deformation Theory .

For the special case that $C={\mathrm{SAlg}}^{\mathrm{op}}$ is the formal dual of simplicial rings over a characteristic 0 ground field, example 8.6 of Stable ∞-Categories shows that this assignment does indeed reproduce the expected notion of modules over (simplicial) rings. So for this case the above general construction reproduces the one considered by Ben-Zvi/Francis/Nadler in their work on geometric ∞-function theory.

But this also serves to show that nothing in their discussion really depends on the choice of site $C={\mathrm{SRing}}^{\mathrm{op}}$. More generally we may notably use simplicial objects in a site for any smooth topos. This way the above general construction yields a notion of quasicoherent $\mathrm{\infty }$-sheaves – of $\mathrm{\infty }$-vector bundles – on objects in any smooth (∞,1)-topos.

Definition

Context

Given a site $C$, the ∞-stack (∞,1)-topos $H:={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ on $C$ may be thought of as modelling the (∞,1)-category of generalized spaces modeled on $C$. With every such (∞,1)-topos comes canonically a notion of cohomology in $H$ and of principal ∞-bundles in $H$ classified by cocycles.

General definition

Write

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

for the (∞,1)-functor that classifies the tangent (∞,1)-category Cartesian fibration $T_C\to C$ of $C$.

For any $X\in H$ we call

$$\mathrm{\infty }{\mathrm{Mod}}_C(X):=[C^{\mathrm{op}},(\mathrm{\infty }\mathrm{,1})\mathrm{Cat}](X,\mathrm{\infty }{\mathrm{Mod}}_C)$$\infty Mod_C(X) := [C^{op},(\infty,1)Cat](X,\infty Mod_C)

the $(\mathrm{\infty }\mathrm{,1})$-category of canonical quasicoherent modules on $C$. An object in this $(\mathrm{\infty }\mathrm{,1})$-category may be thought of as (the cocycle for) a generalized $\mathrm{\infty }$-vector bundle on $X$.

Related concept

Associated vector bundles

If we think of an object $X\in H$ as a base space, then objects in $\mathrm{\infty }\mathrm{Mod}(X)$ may geometrically be tought of as $\mathrm{\infty }$-vector bundles on $X$.

If however an object $A\in H$ is thought of as a coefficient object for cohomology in $H$, then it is more natural to speak of objects in $\mathrm{\infty }\mathrm{Mod}(A)$ as representations of $A$.

For instance if $A=BG$ is the delooping of a group object, then a morphism $BG\to \mathrm{\infty }\mathrm{Mod}$ picks a single $\mathrm{\infty }$-module $V$ equipped with an action of $H$ on it by module homomorphisms. This makes $V$ a representation of $G$.

So in a higher categorical context modules, vector bundles and representations are all different aspects of the same general structure. Accordingly there is an evident categorification of the notion of associated bundle:

For $X,A$ objects in $H$, the ∞-groupoid $H(X,A)$ of $A$-valued cocycles on $X$ is canonically identified with the $\mathrm{\infty }$-groupoid of $A$-principal ∞-bundles on $X$.

For $\rho :A\to \mathrm{\infty }\mathrm{Mod}$ a representation of $A$, we say that the induced morphism

$$\begin{array}{rl}{E}_{\rho }:H(X,A)& \to \mathrm{\infty }\mathrm{Mod}(X)\\ (X\to A)& \mapsto (X\to A\stackrel{\rho }{\to }\mathrm{\infty }\mathrm{Mod})\end{array}$$\begin{aligned} E_\rho : \mathbf{H}(X,A) & \to \infty Mod(X) \\ (X \to A) & \mapsto (X \to A \stackrel{\rho}{\to} \infty Mod) \end{aligned}

sends $A$-principal $\mathrm{\infty }$-bundles to their $\rho$-associated (generalized) $\mathrm{\infty }$-vector bundles.

Flat $\mathrm{\infty }$-vector bundles

For $X\in H$ a space and $A\in H$ a coefficient object, the cohomology of the path ∞-groupoid $\Pi (X)$ of $X$ is the flat differential cohomology $H(\Pi (X),A)$ classifying flat $A$-principal ∞-bundles on $X$.

Accordingly, we think of morphisms

$$\Pi (X)\to \mathrm{\infty }\mathrm{Mod}$$\Pi(X) \to \infty Mod

as (cocycles for) flat $\mathrm{\infty }$-vector bundles on $X$.

Under suitable conditions, as discussed at ∞-Lie differentiation and integration, we have that finite parallel transport is already equivalent to infinitesimal parallel transport, and flat $A$-principal ∞-bundles $\Pi (X)\to A$ are already equivalently encoded by their restriction along the inclusion ${\Pi }^{\inf }(X)\to \Pi (X)$ of the infinitesimal path ∞-groupoid of $X$, and given by morphisms

$${\Pi }^{\inf }(X)\to A$$\Pi^{inf}(X) \to A

that encodes the corresponding flat ∞-Lie algebroid valued differential forms.

Accordingly, we may think of morphisms

$${\Pi }^{\inf }(X)\to \mathrm{\infty }\mathrm{Mod}$$\Pi^{inf}(X) \to \infty Mod

as generalized$\mathrm{\infty }$-vector bundles equipped with an infinitesimal notion of flat connection. In terms of quasicoherent $\mathrm{\infty }$-stacks this are D-modules on $X$.

For notice the decategorification to the classical picture:

• ${\Pi }^{\inf }(X)$ decategorifies to the deRham space of $X$;

• $\mathrm{\infty }\mathrm{Mod}(Y)$ decategorifies to the ordinary (1-categorical) quasicoherent sheaves on $Y$

Hence $\mathrm{\infty }\mathrm{Mod}({\Pi }^{\inf }(X))$ decategorifies to the quasicohefrent sheaves $\mathrm{QC}(\mathrm{dR}(X))$ on the deRham space of $X$, which are D-modules on $X$ (as described there).

Examples

to be filled in

Derived quasicoherent sheaves

to be filled in

In their work on geometric ∞-function theory, David Ben-Zvi, John Francis and David Nadler have studied the generalization of this construction from the site of rings to that of simplicial rings . Again, there is a canonical bifibration $\mathrm{SMod}\to \mathrm{SRing}$ of modules over simplicial rings, classified by a functor $\mathrm{Vect}:\mathrm{SRing}\to (\mathrm{\infty }\mathrm{,1})\mathrm{Cat}$, and this plays the role of the $(\mathrm{\infty }\mathrm{,2})$-sheaf of $\mathrm{\infty }$-vector bundles in this context. One shows in this context that the pull-push of objects in this bifibration along spans in $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}({\mathrm{SRing}}^{\mathrm{op}})$ nicely models integral transforms such as Fourier-Mukai transformations.

Groupoidification

to be filled in

As discussed at geometric function theory, the above geometric ∞-function theory looks in several aspects like a generalization of the situation studied by John Baez, Jim Dolan and Todd Trimble under the term groupoidification. In our terms here, what these authors study is in the context of the site $C=$ Grpd, the pull-push with respect to the codomain fibration $\mathrm{cod}:[I,C]toC$: that which is classified by the functor that sends each object to its overcategory (see geometric function object for more on this).

And for $C=\mathrm{Grpd}$ this differs from the codomain fibration that implicitly underlies the Baez-Dolan situation only in that it assigns stabilized overcateories, instead of overcategories itself. But this should in fact be the step necessary to fully linearize the Baez-Dolan setup, something that these authors propose to emulate by other means.

Thanks to Zoran Skoda and Domenico Fiorenza for discussion here and to David Ben-Zvi and John Francis and Thomas Nikolaus for discussion here about this stuff.