Subsection 3.1.2. (0NZJ) · ScalingStacks

3.1.2. Generators

Now we review notions of what it means for compact objects to generate a stable $\infty$ -category. (See [L3, 17] for more details, and [L2, 5.5.7] for the general setting of presentable $\infty$ -categories.)

0NWK

Definition 3.7. A stable category $\mathcal{C}$ is said to be compactly generated if there is a small $\infty$ -category $\mathcal{C}^{\circ}$ of compact objects $C_{i}\in\mathcal{C}$ whose right orthogonal vanishes: if $M\in\mathcal{C}$ satisfies $\operatorname{\Hom}_{\mathcal{C}}(C_{i},M)\simeq 0$ , for all $i$ , then $M\simeq 0$ .

As explained in [L3, Remark 17.3], whether a stable $\infty$ -category is compactly generated can be studied completely in the underlying homotopy category. In particular, the notion for stable $\infty$ -categories is compatible with that for triangulated categories.

0NWL

Example 3.8. For a commutative derived ring $k$ , the stable $\infty$ -category $\Mod_{k}$ of $k$ -modules is compactly generated. In fact, it is generated by the free module $k$ itself.

On the one hand, for a stable small $\infty$ -category $\mathcal{C}^{\circ}$ , the inductive limit $\mathcal{C}=\operatorname{Ind}(\mathcal{C}^{\circ})$ is a compactly generated stable presentable $\infty$ -category. Furthermore (see [L2, 5.3.4]), if $\mathcal{C}^{\circ}$ is closed under finite colimits and idempotent complete, then it can be recovered as the compact objects of $\mathcal{C}$ . In particular, we have that $\mathcal{C}$ is the Ind-category of its compact objects $\mathcal{C}^{\circ}$ .

On the other hand, given a stable $\infty$ -category $\mathcal{C}$ with a small full $\infty$ -subcategory $\mathcal{C}^{\circ}$ of compact generators, one can recover all compact objects of $\mathcal{C}$ by a result of Neeman [N1] (see also [L2, Proposition 5.3.4.17]): the compact objects are precisely direct summands of the objects of the smallest stable $\infty$ -subcategory $\mathcal{C}_{s}$ containing $\mathcal{C}^{\circ}$ (that is, they are direct summands of finite iterated extensions of objects of $\mathcal{C}^{\circ}$ ). In particular, if $\mathcal{C}^{\circ}$ is stable and idempotent complete, then it consists precisely of the compact objects of $\mathcal{C}$ .

If we further assume that $\mathcal{C}$ is a presentable stable $\infty$ -category $\mathcal{C}$ with a small full $\infty$ -subcategory $\mathcal{C}^{\circ}$ of compact generators, then a theorem of Schwede and Shipley [SSh] guarantees that we can recover $\mathcal{C}$ as the cocompletion of $\mathcal{C}_{s}$ (see [L4, 4.4] for the $\infty$ -categorical version, and [Ke] for the differential graded version). In other words, we recover $\mathcal{C}$ by passing to the category of colimit preserving $k$ -linear functors to $k$ -modules

\mathcal{C}\simeq\Fun(\mathcal{C}_{s}^{\rm op},\Mod_{k}).

In particular, we now can check that our notion of perfect stack is equivalent to more familiar assumptions on a symmetric monoidal $\infty$ -category.

0NWM

Proposition 3.9. For a derived stack $X$ with affine diagonal, the following are equivalent:

(1)

$X$ is perfect.
(2)

$\qc(X)$ is compactly generated, and its compact and dualizable objects coincide.

0NWN

Proof. If $X$ is perfect, so that $\qc(X)=\operatorname{Ind}\operatorname{Perf}(X)$ , we claim that compact and dualizable objects agree, and hence compact objects generate, so that (1) implies (2).

To see the claim, it suffices to show that the full $\infty$ -subcategory of dualizable objects of $\qc(X)$ is idempotent complete (since it is also stable). Since $\qc(X)$ is idempotent complete, this is equivalent to showing that dualizable objects are closed under retracts. However, for a retract $N$ of a dualizable object $M$ one can explicitly write down the unit and trace maps for $N\otimes{\mathcal{H}om}(N,\mathcal{O}_{X})$ and confirm the necessary conditions. We leave this to the reader.

Conversely, if $\qc(X)$ is compactly generated, then $\qc(X)$ is the Ind-category of its compact objects, and hence by assumption also the Ind-category of its dualizable objects, so that (2) implies (1). ∎

David Ben-Zvi, John Francis, David Nadler

Original source: arXiv:0805.0157v5