Subsection 1.2. (0NRF) · ScalingStacks

1.2. Morita theory

The main technical device in our proofs of theorems 1.1 and 1.3 is the Morita theory of stable categories and spectral categories. In particular, we prove theorem 1.3 by using a comparison result between the theory of small spectral categories (see §2.1) and the theory of small stable $\infty$ -categories to rigidify questions about the algebraic $K$ -theory of $\infty$ -categories to corresponding questions in the (classical) Waldhausen $K$ -theory of Waldhausen categories.

The category $\Cat_{\mathcal{S}}$ of small spectral categories carries a Quillen model category structure in which the weak equivalences are the DK-equivalences, i.e., the functors that are fully faithful and essentially surjective up to weak homotopy equivalence; see [74] (reprised below in theorem 2.2). As a consequence, we can form the associated $\infty$ -category $(\Cat_{\mathcal{S}})_{\infty}$ of small spectral categories.

A spectral functor $F\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is called a triangulated equivalence if it induces a weak equivalence on the triangulated closures of ${\mathcal{A}}$ and ${\mathcal{B}}$ , and it is called a Morita equivalence if it induces a weak equivalence on the thick closures of ${\mathcal{A}}$ and ${\mathcal{B}}$ ; see definition 2.7. Our comparison result, which can be regarded as a generalization of the Morita theory of [69], is the following.

0NJ4

Theorem 1.10. (see theorems 4.22 and 4.23) The accessible localization of $(\Cat_{\mathcal{S}})_{\infty}$ along the triangulated equivalences is equivalent to $\Cat_{\infty}^{\ex}$ , and the (further) localization of $(\Cat_{\mathcal{S}})_{\infty}$ along the Morita equivalences is equivalent to $\Cat_{\infty}^{\perf}$ .

We use this comparison result to deduce structural properties of the categories $\Cat_{\infty}^{\ex}$ and $\Cat_{\infty}^{\perf}$ , notably that they are compactly generated, complete, and cocomplete; see corollary 4.25. Furthermore, we prove theorem 1.3 by using theorem 1.10 to lift split-exact sequences of small $\infty$ -categories to split-exact sequences of small spectral categories so we can take the $K$ -theory in the setting of Waldhausen categories. More generally, this comparison result explains the relationship between the classical versions of algebraic $K$ -theory (and topological Hochschild and cyclic homology) and the $\infty$ -categorical versions. We believe that theorem 1.10 is of independent interest and expect it will find applications in the future. For instance, this theorem provides clean and concise proofs of the main theorems of Toën’s work [80] on internal $\hom$ objects in the category of dg-categories and its (previously unknown) extension to the context of spectral categories.

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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