∞-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 \infty-vector bundle.



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

and on

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 \infty-vector bundles on spaces in any smooth (∞,1)-topos. There is an evident notion of \infty-vector bundles associated to any principal ∞-bundle and from the theory of differential nonabelian cohomology we obtain a notion of smooth \infty-vector bundles with connection.


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 XX is nothing but a morphism of stacks XModX \mapsto Mod into the canonical stack Mod:SpecAAModMod : Spec A \to A Mod of modules. (For details on this see the discussion at quasicoherent sheaf.)

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

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

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 ) \,.

The Cartesian fibration classified by this is the tangent (∞,1)-category T CCT_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=SAlg opC = SAlg^{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=SRing opC = SRing^{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 \infty-sheaves – of \infty-vector bundles – on objects in any smooth (∞,1)-topos.



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

General definition


Mod C:C opStab(,1)Cat(,1)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 CCT_C \to C of CC.

For any XHX \in \mathbf{H} we call

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

the (,1)(\infty,1)-category of canonical quasicoherent modules on CC. An object in this (,1)(\infty,1)-category may be thought of as (the cocycle for) a generalized \infty-vector bundle on XX.

Associated vector bundles

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

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

For instance if A=BGA = \mathbf{B}G is the delooping of a group object, then a morphism BGMod\mathbf{B}G \to \infty Mod picks a single \infty-module VV equipped with an action of HH on it by module homomorphisms. This makes VV a representation of GG.

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,AX,A objects in H\mathbf{H}, the ∞-groupoid H(X,A)\mathbf{H}(X,A) of AA-valued cocycles on XX is canonically identified with the \infty-groupoid of AA-principal ∞-bundles on XX.

For ρ:AMod\rho : A \to \infty Mod a representation of AA, we say that the induced morphism

E ρ:H(X,A) Mod(X) (XA) (XAρMod) \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 AA-principal \infty-bundles to their ρ\rho-associated (generalized) \infty-vector bundles.

Flat \infty-vector bundles

For XHX \in \mathbf{H} a space and AHA \in \mathbf{H} a coefficient object, the cohomology of the path ∞-groupoid Π(X)\Pi(X) of XX is the flat differential cohomology H(Π(X),A)\mathbf{H}(\Pi(X),A) classifying flat AA-principal ∞-bundles on XX.

Accordingly, we think of morphisms

Π(X)Mod \Pi(X) \to \infty Mod

as (cocycles for) flat \infty-vector bundles on XX.

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 AA-principal ∞-bundles Π(X)A\Pi(X) \to A are already equivalently encoded by their restriction along the inclusion Π inf(X)Π(X)\Pi^{inf}(X) \to \Pi(X) of the infinitesimal path ∞-groupoid? of XX, and given by morphisms

Π inf(X)A \Pi^{inf}(X) \to A

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

Accordingly, we may think of morphisms

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

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

For notice the decategorification to the classical picture:

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


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 SModSRingSMod \to SRing of modules over simplicial rings, classified by a functor Vect:SRing(,1)CatVect : SRing \to (\infty,1)Cat, and this plays the role of the (,2)(\infty,2)-sheaf of \infty-vector bundles in this context. One shows in this context that the pull-push of objects in this bifibration along spans in H=Sh (,1)(SRing op)\mathbf{H} = Sh_{(\infty,1)}(SRing^{op}) nicely models integral transforms such as Fourier-Mukai transformations.


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=C = Grpd, the pull-push with respect to the codomain fibration cod:[I,C]toCcod : [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=GrpdC = 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.

Revised on May 29, 2012 22:04:00 by Andrew Stacey (