### Quasicoherent ∞ -Stacks

#### Posted by Urs Schreiber

This is to tie up a loose end in our discussion of Ben-Zvi/Francis/Nadler geometric $\infty$-function theory and Baez/Dolan/Trimble “groupoidification”.

Using a central observation in Lurie’s *Deformation Theory* one can see how both these approaches are special examples of a very general-abstract-nonsense
theory of $\infty$-linear algebra on $\infty$-vector bundles and
$\infty$-quasicoherent sheaves in arbitrary (“derived”) $\infty$-stack
$(\infty,1)$-toposes:

the notion of module is controled by the tangent $(\infty,1)$-category of the underlying site, that of quasicoherent module/$\infty$-vector bundle simply by homs into its classifying $\infty$-functor.

For Ben-Zvi/Francis/Nadler this shows manifestly, I think, that nothing in their article really depends on the fact that the underlying site is chosen to be that of duals of simplicial rings. It could be any $(\infty,1)$-site. For Baez/Dolan/Trimble it suggests the right way to fix the linearity: their setup is that controlled by the tangent $(\infty,1)$-category of $\infty Grpd$ itself and where they (secretly) use the codomain bifibration over this, one should use the fiberwise stabilized codomain fibration.

See $\infty$-vector bundle for a discussion of what I have in mind.

What I am saying here is likely very obvious to somebody out there. I have my suspicions. But it looks like such a nice fundamental fact, that this deserves to be highlighted, and be it in a blog entry. It is just a matter of putting 1 and 1 together. The two central observations are this:

In our journal club discussion I had remarked that the definition $QC(X)$ of derived quasicoherent sheaves on a derived $\infty$-stack $X$ that BenZvi/Francis/Nadler use is best thought of as $hom(X,QC)$ in the $(\infty,1)$-category of $(\infty,1)$-category-valued $\infty$-stacks. That this is the way to think about quasicoherent sheaves must be an old hat to some people but is rarely highlighted in the literature.

The only place I know of presently where this is made fully explicit is the $n$Lab entry on quasicoherent sheaves. As discussed there, at least Kontsevich and Rosenberg have made this almost fully explicit – they say this in side remark 1.1.5 here, in the dual picture where category-valued presheaves are replaced by fibered categories.

The other crucial insight is Lurie’s basic idea from *Deformation Theory*.
This is amazingly elegant, have a look, if you haven’t yet. Lurie shows that
for $C$ any $(\infty,1)$-category, whose objects we here think of as formal duals of
test spaces, the tangent $(\infty,1)$-category fibration $T_C \to C$ is to
be thought of as the fibration of modules over the objects of $C$, generalizing
the canonical fibration $Mod \to Ring$ that underlies the classical theory
of monadic descent. But strikingly: $T_C$ is effectively nothing but the codomain fibration over $C$ – it differs from that only in that
all fibers are stabilized, i.e. linearized in the fully abstract category-theoretic sense.
In particular, this says that the Ben-Zvi/Francis/Nadler $\infty$-stack of
derived quasicoherent sheaves $QC : SRing \to (\infty,1)Cat$ is equivalently simply
given by assigning $Spec A \mapsto Stab(SRing/A)$ – to any simplicial ring its
stabilized overcategory. (This is stated in example 8.6 on page 24 in *Stable $\infty$-Categories*.)

I had remarked here in our journal club discussion that Baez/Dolan/Trimble “groupoidification” is like Ben-Zvi/Francis/Nadler theory but with $QC$ replaced by the assignment of overcategories. Now in the light of Lurie’s observation this makes everything fall into place: Baez/Dolan/Trimble groupoidification may be thought of as a shadow of the geometric $\infty$-function theory induced from the tangent $(\infty,1)$-category over $\infty Grpd$ itself: instead of pull-pushing bundles of groupoids as they do, in the full theory one would pull-push bundles of abelian $\infty$-groupoids (and more generally: possibly non-connective spectra).

You may remember my motivation for coming to grips with this conceptual framework: for applications in the physics of gauge fields we need smooth differential cohomology and $\infty$-Lie theory. This doesn’t really take place in $\infty$-stacks over simplicial rings (only a small part of it does) and it doesn’t take place in $\infty$-stacks over just $\infty Grpd$ (though important toy models do): it takes place in $\infty$-stacks over simplicial $C^\infty$-rings and generally over simplicial objects in sites for smooth toposes. Clearly we want Ben-Zvi/Francis/Nadler geometric $\infty$-function theory generalized to this context. With $QC$ in hand in this contex, we immediately have $\infty$-representations of $\infty$-Lie groups, associated $\infty$-vector bundles and their pull-push and monadic descent.

And with the above it is clear what one needs to look at: just take the $\infty$-stack of smooth generalized $\infty$-vector bundles to be $QC : Spec A \mapsto Stab(SC^\infty Ring/A)$. And that’s it.

More at $\infty$-vector bundle.

## Re: Quasicoherent infty -Stacks

The link in

Using a central observation in Lurie’s Deformation Theory

seems to produce only a cover page

Remind me how to use it? i.e how to get a coherent! single document

hopefully not by clicking on each item in the Contents separately