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 $X$ is nothing but a morphism of stacks $X \mapsto Mod$ into the canonical stack $Mod : Spec A \to A Mod$ of modules. (For details on this see the discussion at quasicoherent sheaf.)
Of this there is an obvious categorification: for $\infty Mod$ 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 $X \to \infty Mod$ as defining a “quasicoherent” $\infty$-module on $X$, with respect to the notion of module encoded by $\infty Mod$.
But moreover, after passage to (∞,1)-categories, a little miracle happens: there is now a canonical notion of $\infty Mod$ for any given site $C$: if we regard the objects of $C$ as test spaces and hence the objects of the oppposite (∞,1)-category $C^{op}$ as function rings on test spaces, then the $(\infty,1)$-category $\infty Mod(R)$ of modules over the function ring $R$ may be identitfied simply with the stabilization of the overcategory of $C$ over $R$:
or directly in terms of test spaces $U$:
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 $C = SAlg^{op}$ 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 $C = SRing^{op}$. 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 $C$, the ∞-stack (∞,1)-topos $\mathbf{H} := Sh_{(\infty,1)}(C)$ 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 $\mathbf{H}$ and of principal ∞-bundles in $\mathbf{H}$ classified by cocycles.
Write
for the (∞,1)-functor that classifies the tangent (∞,1)-category Cartesian fibration $T_C \to C$ of $C$.
For any $X \in \mathbf{H}$ we call
the $(\infty,1)$-category of canonical quasicoherent modules on $C$. An object in this $(\infty,1)$-category may be thought of as (the cocycle for) a generalized $\infty$-vector bundle on $X$.
If we think of an object $X \in \mathbf{H}$ as a base space, then objects in $\infty Mod(X)$ may geometrically be tought of as $\infty$-vector bundles on $X$.
If however an object $A \in \mathbf{H}$ is thought of as a coefficient object for cohomology in $\mathbf{H}$, then it is more natural to speak of objects in $\infty Mod(A)$ as representations of $A$.
For instance if $A = \mathbf{B}G$ is the delooping of a group object, then a morphism $\mathbf{B}G \to \infty Mod$ picks a single $\infty$-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 $\mathbf{H}$, the ∞-groupoid $\mathbf{H}(X,A)$ of $A$-valued cocycles on $X$ is canonically identified with the $\infty$-groupoid of $A$-principal ∞-bundles on $X$.
For $\rho : A \to \infty Mod$ a representation of $A$, we say that the induced morphism
sends $A$-principal $\infty$-bundles to their $\rho$-associated (generalized) $\infty$-vector bundles.
For $X \in \mathbf{H}$ a space and $A \in \mathbf{H}$ a coefficient object, the cohomology of the path ∞-groupoid $\Pi(X)$ of $X$ is the flat differential cohomology $\mathbf{H}(\Pi(X),A)$ classifying flat $A$-principal ∞-bundles on $X$.
Accordingly, we think of morphisms
as (cocycles for) flat $\infty$-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
that encodes the corresponding flat ∞-Lie algebroid valued differential forms.
Accordingly, we may think of morphisms
as generalized$\infty$-vector bundles equipped with an infinitesimal notion of flat connection. In terms of quasicoherent $\infty$-stacks this are D-modules on $X$.
For notice the decategorification to the classical picture:
$\Pi^{inf}(X)$ decategorifies to the deRham space of $X$;
$\infty Mod(Y)$ decategorifies to the ordinary (1-categorical) quasicoherent sheaves on $Y$
Hence $\infty Mod(\Pi^{inf}(X))$ decategorifies to the quasicohefrent sheaves $QC(dR(X))$ on the deRham space of $X$, which are D-modules on $X$ (as described there).
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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 $SMod \to SRing$ of modules over simplicial rings, classified by a functor $Vect : SRing \to (\infty,1)Cat$, and this plays the role of the $(\infty,2)$-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 $\mathbf{H} = Sh_{(\infty,1)}(SRing^{op})$ nicely models integral transforms such as Fourier-Mukai transformations.
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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 $cod : [I,C] \toC$: that which is classified by the functor that sends each object to its overcategory (see geometric function object for more on this).
And for $C = Grpd$ 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.