Subsection 1.3. (0NRG) · ScalingStacks

1.3. Symmetric monoidal structure and dualizable objects

The category $\Cat_{\infty}^{\perf}$ is a symmetric monoidal $\infty$ -category (in the sense of [53, §2]), in which the tensor product $\widehat{\otimes}$ is characterized by the property that maps out of ${\mathcal{A}}\widehat{\otimes}{\mathcal{B}}$ correspond to maps out of the product ${\mathcal{A}}\times{\mathcal{B}}$ which preserve colimits in each variable; see section 3 for a discussion of this structure, following the work of Lurie [53] and Ben-Zvi, Francis, and Nadler [8]. We will reserve a careful study of the structure of ${\mathcal{M}}_{\mathrm{add}}$ and ${\mathcal{M}}_{\mathrm{loc}}$ as symmetric monoidal $\infty$ -categories for the forthcoming paper [9]. However, in order to carry out the extension of the non-connective co-representability theorem described in remark 1.8, we study the theory of dualizable objects in $\Cat_{\infty}^{\perf}$ , using the theory of [53, §4.2.5].

In analogy with the situation for dg-categories [21, §4], we obtain a characterization of the dualizable objects in $\Cat_{\infty}^{\perf}$ as the smooth and proper objects. We define these notions as follows. Implicit in the comparison between small stable $\infty$ -categories and spectral categories of theorem 1.10 is the fact that for objects $a$ and $b$ in a small stable $\infty$ -category ${\mathcal{A}}$ there exists a natural mapping spectrum ${\mathcal{A}}(a,b)$ (see definition 2.15). Using this fact, we say that a small stable $\infty$ -category ${\mathcal{A}}$ is proper if, for all pairs of objects $a$ and $b$ of ${\mathcal{A}}$ , the mapping spectrum ${\mathcal{A}}(a,b)$ is compact. We say that a small stable $\infty$ -category ${\mathcal{A}}$ is smooth if it is perfect as an ${\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{A}}$ -module. (Here we use the fact that any small stable $\infty$ -category can be regarded as a bimodule over itself.) We then have the following theorem characterizing the dualizable objects in these terms:

0NJ5

Theorem 1.11. (see theorem 3.7) An idempotent-complete small stable $\infty$ -category ${\mathcal{A}}$ is dualizable (as an object of the symmetric monoidal $\infty$ -category $\Cat_{\infty}^{\perf}$ ) if and only if ${\mathcal{A}}$ is smooth and proper. Moreover, the dual of a dualizable idempotent-complete small stable $\infty$ -category ${\mathcal{A}}$ is the opposite $\infty$ -category ${\mathcal{A}}^{\op}$ .

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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