ScalingStacks

For an arbitrary derived stack $X$, the $\infty$-category $\qc(X)$ is difficult to control algebraically. For example, it may contain large objects that are impossible to construct in terms of concrete, locally-finite objects.

To get a handle on $\qc(X)$, we need to know that it has a small $\infty$-subcategory $\qc(X)^{\circ}$ of “generators” which are “finite” in an appropriate sense. There are two common notions of when a small subcategory $\mathcal{C}^{\circ}$ generates a category $\mathcal{C}$, and they have natural $\infty$-analogues. On the one hand, we could ask that $\mathcal{C}$ be the inductive limit or ind-category $\operatorname{Ind}\mathcal{C}^{\circ}$ (i.e., that $\mathcal{C}$ is freely generated from $\mathcal{C}^{\circ}$ by taking inductive limits), and on the other hand, we could ask that in $\mathcal{C}$ the right orthogonal of $\mathcal{C}^{\circ}$ vanishes. When $\mathcal{C}$ is $\qc(X)$, there are three common notions of which objects may be considered finite: perfect objects, dualizable objects, and compact objects, which refer respectively to the geometry, monoidal structure, and categorical structure of $\qc(X)$. We review all of these notions and the relations between them in Section 3.1 (in particular, perfect and dualizable objects always coincide).

To have a tractable and broadly applicable class of derived stacks, we introduce the notion of a perfect stack. By definition, an object $M$ of $\qc(X)$ is a perfect complex if locally for any affine $U\to X$ its restriction to $U$ is a perfect module (finite complex of projective modules). Equivalently, $M$ is dualizable with respect to the monoidal structure on $\qc(X)$. A derived stack $X$ is said to be perfect if it has affine diagonal and the $\infty$-category $\qc(X)$ is the inductive limit

of the full $\infty$-subcategory $\operatorname{Perf}(X)$ of perfect complexes. A morphism $X\to Y$ is said to be perfect if its fibers $X\times_{Y}U$ over affines $U\to Y$ are perfect.

On a perfect stack $X$, compact objects of $\qc(X)$ are the same thing as perfect complexes (which in turn are always the same thing as dualizable objects). In fact, we have the following alternative formulation: a derived stack $X$ is perfect if it has affine diagonal, $\qc(X)$ is compactly generated (that is, there is no right orthogonal to the compact objects), and compact objects and dualizable objects coincide.

In Section 3.2, we establish base change and the projection formula for perfect morphisms following arguments in Lurie’s thesis [L1] (in fact, the arguments here only use that the structure sheaf is relatively compact).

In Section 3.3, we show that the class of perfect stacks is very broad. We show that the following are all perfect stacks:

1. (1)

   Quasi-compact derived schemes with affine diagonal (following arguments of [N2]).

2. (2)

   The total space of a quasi-projective morphism over a perfect base.

3. (3)

   In characteristic zero, the quotient $X/G$ of a quasi-projective derived scheme $X$ by a linear action of an affine group $G$.

4. (4)

   The quotient $X/G$ of a perfect stack $X$ by a finite affine group scheme in “good” characteristics for $G$.

5. (5)

   The mapping stack $X^{\Sigma}=\Map(\Sigma,X)$, for a perfect stack $X$ and a finite simplicial set $\Sigma$.

6. (6)

   Fiber products of perfect stacks.

We also show that any morphism $X\to Y$ between perfect stacks is itself perfect.

Though the above examples show that perfect stacks cover a broad array of spaces of interest, it is worth pointing out that there are many commonly arising derived stacks that are imperfect. Since categories of quasi-coherent sheaves are usually compactly generated, the typical reason for $X$ to be imperfect is that the structure sheaf $\mathcal{O}_{X}$ (which is always dualizable) fails to be compact (or equivalently, the global sections functor fails to preserve colimits). This can occur if the cohomology $\Gamma(X,\mathcal{O}_{X})$ is too big such as for (1) the classifying space of a finite group in modular characteristic (for the simplest example, one can take $B\mathbb{Z}/2\mathbb{Z}$ when $2$ is not invertible), (2) the classifying space of a topological group such as $S^{1}$, or (2) an ind-scheme such as the formal disc $\operatorname{Spf}k[[t]]$. In the opposite direction, categories of ${\mathcal{D}}$-modules on smooth schemes (which can be considered as quasi-coherent sheaves on the corresponding de Rham stacks) have compact unit, but also many compact objects (such as ${\mathcal{D}}$ itself) which are not dualizable.

For $X,X^{\prime}$ ordinary schemes over a ring $k$, a theorem of Toën [To1] identifies the dg category of $k$-linear continuous (that is, colimit preserving) functors with the dg category of integral kernels

For dg categories of perfect (equivalently, bounded coherent) complexes on smooth projective varieties, an analogous result was proved by Bondal, Larsen and Lunts [BLL] as well as by Toën [To1] (generalizing Orlov’s theorem [O] characterizing equivalences as Fourier-Mukai transforms).

We’ve collected our main technical results in the following generalization. In the statement, the tensors and functors of $\infty$-categories of quasi-coherent sheaves are calculated in the symmetric monoidal $\infty$-category $\mathcal{P}r^{\rm L}$ of presentable $\infty$-categories with morphisms left adjoints (as developed in [L4, 4] and [L5, 5], see Section 2 for a precise summary). The tensors and functors of $\infty$-categories of perfect complexes are calculated in the symmetric monoidal $\infty$-category ${st}$ of $k$-linear idempotent complete stable small $\infty$-categories (as developed in Section 4.1 below).

Our arguments in Section 4 first establish the pair of assertions about tensors, then deduce sufficient duality to conclude the pair of assertions about functors.

A basic operation on associative algebras is the calculation of their center. The derived version of the center of an associative algebra $A$ is the Hochschild cochain complex (or simply the Hochschild cohomology), which calculates the derived endomorphisms of $A$ as an $A$-bimodule. Another basic operation is the calculation of the universal trace (i.e., the universal target for a map out of $A$ coequalizing left and right multiplication). The derived version of the universal trace is the Hochschild chain complex (or the Hochschild homology), which calculates the derived tensor product of $A$ with itself as an $A$-bimodule.

In Section 5.1, we extend the notion of Hochschild homology and cohomology to associative (or $\mathcal{E}_{1}$-)algebra objects in arbitrary closed symmetric monoidal $\infty$-categories. (As with any structure in an $\infty$-category, an associative multiplication, or $\mathcal{E}_{1}$-structure, is a homotopy coherent notion.) In the case of chain complexes, we recover the usual Hochschild chain and cochain complexes. In the case of spectra, we recover topological Hochschild homology and cohomology.

We show in particular that $\hh^{*}(A)$ and $\hh_{*}(A)$ are calculated in this generality by a version of the usual Hochschild complexes, the cyclic bar construction.

We apply this definition in the following setting. We will take $\mathcal{S}$ to be the $\infty$-category ${\mathcal{P}r}^{\rm L}$ of presentable $\infty$-categories with morphisms left adjoints. Then an associative algebra object in $\mathcal{P}r^{\rm L}$ is a monoidal presentable $\infty$-category $\mathcal{C}$. Thus we have the notion of its center (or Hochschild cohomology category) and trace (or Hochschild homology category)

These are again presentable $\infty$-categories, or in other words, objects of the $\infty$-category $\mathcal{P}r^{\rm L}$. The center $\mathcal{Z}(\mathcal{C})$ comes equipped with a universal central functor to $\mathcal{C}$, and the trace $\mathcal{T}r(\mathcal{C})$ receives a universal trace functor from $\mathcal{C}$.

Now let us return to a geometric setting and consider a perfect stack $X$ and the presentable $\infty$-category $\qc(X)$. Since $\qc(X)$ is symmetric monoidal, it defines a commutative (or $\mathcal{E}_{\infty}$-)algebra object in $\mathcal{P}r^{\rm L}$, and so in particular, an associative algebra object.

To calculate the center and trace of $\qc(X)$, we introduce the loop space

where the derived fiber product is along two copies of the diagonal map.

For example, when $X$ is an ordinary smooth scheme over a field of characteristic zero, the loop space $\mathcal{L}X$ is the total space $T_{X}[-1]=\Spec\operatorname{Sym}\Omega_{X}[1]$ of the shifted tangent bundle of $X$. When $X$ is the classifying space $BG$ of a group $G$, the loop space $\mathcal{L}BG$ is the adjoint quotient $G/G$.

In Section 5.1, as a corollary of our main technical results, we obtain the following.

Note that the theorem in particular identifies the Hochschild homology and cohomology objects associated to the monoidal $\infty$-category $\qc(X)$. While such an identification may initially appear surprising, it is a natural consequence of the self-duality of $\qc(X)$ over $\qc(X\times X)$ which in turn is a simple consequence of Theorem 1.2.

In Section 5.3, we also discuss a generalization of the theorem to $\mathcal{E}_{n}$-centers and traces when $n>1$. First, we introduce the notion of center and trace for an $\mathcal{E}_{n}$-algebra object in $\mathcal{P}r^{\rm L}$, for any $n$. Since $\qc(X)$ is an $\mathcal{E}_{\infty}$-algebra object in $\mathcal{P}r^{\rm L}$, it is also an $\mathcal{E}_{n}$-algebra object, for any $n$.

We show that the $\mathcal{E}_{n}$-center and trace of $\qc(X)$ are equivalent to the $\infty$-category of quasi-coherent sheaves on the derived mapping space

For example, when $X$ is an ordinary smooth scheme over a field of characteristic zero, the $n$-sphere space $X^{S^{n}}$ is the total space $T_{X}[-n]=\Spec\operatorname{Sym}\Omega_{X}[n]$ of the shifted tangent bundle of $X$. In particular, as we vary $n$, the $\mathcal{E}_{n}$-centers and traces of $\qc(X)$ differ from $\qc(X)$, though they all have $t$-structures with heart the abelian category of quasi-coherent sheaves on $X$.

When $X$ is the classifying space $BG$ of a group $G$, the $n$-sphere space $BG^{S^{n}}$ can be interpreted as the derived stack ${\mathcal{L}oc}_{G}(S^{n})$ of $G$-local systems on $S^{n}$. In particular, when $n=2$, the $\mathcal{E}_{2}$-center and trace of $\qc(BG)$ is the $\infty$-category $\qc({\mathcal{L}oc}_{G}(S^{2}))$ which appears in the Geometric Langlands program (see for example, the work of Bezrukavnikov and Finkelberg [BeF] who identify $\qc({\mathcal{L}oc}_{G}(S^{2}))$ with the derived Satake (or spherical Hecke) category of arc-group equivariant constructible sheaves on the affine Grassmannian for the dual group).

Let $X\to Y$ be a map of perfect stacks. Our main technical result gives an identification of (not necessarily symmetric) monoidal $\infty$-categories

where the left hand side is equipped with convolution and the right hand side with the composition of functors. In other words, we have an identification of associative algebra objects in the $\infty$-category ${\mathcal{P}r}^{\rm L}$ of presentable $\infty$-categories with morphisms left adjoints.

In Section 5.2, we calculate the center of $\qc(X\times_{Y}X)$ in the following form. Recall that $\mathcal{L}Y$ denotes the derived loop space of $Y$.

When $p:X\to Y$ is also proper with invertible dualizing complex, we deduce an analogous statement identifying the trace of $\qc(X\times_{Y}X)$ with $\qc(\mathcal{L}Y)$, conditional on the validity of Grothendieck duality in the derived setting.

As we discuss in Section 6, our results may be viewed from the perspective of topological field theory.

The $\infty$-category $\qc(\mathcal{L}X)$ (the Drinfeld center $\mathcal{Z}(\qc(X))$) carries a rich collection of operations generalizing the braided tensor structure on modules for the classical Drinfeld double. Namely, for any cobordism $\Sigma$ between disjoint unions of circles $(S^{1})^{\coprod m}$ and $(S^{1})^{\coprod n}$, we obtain restriction maps between the corresponding mapping stacks

Pullback and pushforward along this correspondence defines a functor

which is compatible with composition of cobordisms. In particular, we obtain a map from the configuration space of $m$ small disks in the standard disk to functors

This equips $\qc(\mathcal{L}X)$, and hence the Drinfeld center $\mathcal{Z}(\qc(X))$, with the structure of a (framed) $\mathcal{E}_{2}$-category. This establishes the categorified (cyclic) Deligne conjecture in the current geometric setting.

In particular for $X=BG$, $X^{\Sigma}$ is the derived moduli stack of $G$-local systems on $\Sigma$ and we obtain a topological gauge theory.

More generally, we have the following corollary of our main results:

One can view this corollary as providing for a TFT over finite simplicial sets. We assign to such a simplicial set $U$ the $\infty$-category $\qc(X^{U})$, and for any diagram of finite simplicial sets

(for example a cobordism of manifolds) we obtain a correspondence of mapping stacks

and hence by pullback and pushforward a functor

satisfying composition laws with respect to gluings.

In particular, since $\qc(X^{S^{n}})$ is equivalent to the $\mathcal{E}_{n}$-center $\mathcal{Z}_{\mathcal{E}_{n}}(\qc(X))$, we obtain an action of the (framed) $\mathcal{E}_{n+1}$-operad on $\mathcal{Z}_{\mathcal{E}_{n}}(\qc(X))$. This establishes the categorified (cyclic) Kontsevich conjecture in the current geometric setting.

This paper relies heavily on the work of Jacob Lurie, whom the authors thank for many helpful discussions and for his generosity in sharing his knowledge (and in particular for the suggestion of how to describe the higher $\mathcal{E}_{n}$-centers of $\qc(X)$ geometrically). DBZ and DN would like to thank Bertrand Toën for educating them in derived algebraic geometry and patiently explaining many arguments. They would also like to thank Victor Ostrik for sharing his understanding of Drinfeld centers and their applications to Hecke categories. JF thanks his advisor, Michael Hopkins, for discussions on the topological field theory aspects of this paper, as well as for his support and guidance. Many thanks to Andrew Blumberg and Amnon Neeman for detailed comments on early drafts. Finally, we would also like to thank the anonymous referees for their valuable feedback.

DBZ is partially supported by NSF CAREER grant DMS-0449830. JF was supported by an NSF Graduate Research Fellowship. DN is partially supported by NSF grant DMS-0600909 and a Sloan Research Fellowship. Work on this paper has taken place at the Aspen Center for Physics and the IAS (supported by NSF grant DMS-0635607), whom the authors thank for providing support and stimulating environments.