Subsection 10.1. (0NSM) · ScalingStacks

10.1. $THH$ as a localization invariant

We begin by observing that $THH$ provides a localizing invariant of small stable $\infty$ -categories. Although it is possible to do this directly in the setting of $\infty$ -categories (see for instance the more general discussion of topological chiral homology in [53, §5.3] or the constructions outlined in [8, 5.1.1]), we use existing constructions in the setting of spectral categories in order to ease technical difficulties that arise in the subsequent construction of $TR$ and $TC$ . Our basic sources for this material are [10] and [11].

Recall that for a small spectral category ${\mathcal{C}}$ we can define $THH({\mathcal{C}})$ in terms of the Hochschild-Mitchell cyclic nerve for spectral categories [10, §3]. The cyclic nerve is defined as the simplicial object

N^{\cyc}_{q}{\mathcal{C}}=\bigvee{\mathcal{C}}(c_{q-1},c_{q})\wedge\dotsb\wedge{\mathcal{C}}(c_{0},c_{1})\wedge{\mathcal{C}}(c_{q},c_{0}),

where the sum is over the $(q+1)$ -tuples $(c_{0},\dotsc,c_{q})$ of objects of ${\mathcal{C}}$ . This becomes a simplicial object using the usual cyclic bar construction face and degeneracy maps: The unit maps of ${\mathcal{C}}$ induce the degeneracy maps, and the composition maps in ${\mathcal{C}}$ (along with the twist map at the end) induce the face maps. We denote the geometric realization as $N^{\cyc}{\mathcal{C}}$ .

The spectrum $N^{\cyc}{\mathcal{C}}$ has the correct homotopy type only when ${\mathcal{C}}$ has cofibrant mapping spectra [10, 3.1]. Since the cofibrant objects in the Morita model structure on small spectral categories reviewed in theorem 2.2 have cofibrant mapping spectra [74, 4.18], we can define the functor

THH:=N^{\cyc}\circ Q\colon\Cat_{\mathcal{S}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{S}

where $Q$ denotes the cofibrant replacement functor in the Morita model structure on $\Cat_{\mathcal{S}}$ . Since this construction preserves Morita equivalences [10, 5.12] (and in fact DK-equivalences [10, 5.9]) the functor descends to the level of $\infty$ -categories.

0NQV

Lemma 10.1. The functor

THH\colon\Cat_{\mathcal{S}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{S}

induces a functor of $\infty$ -categories

THH\colon\Cat_{\infty}^{\ex}\simeq\mathrm{N}((\Cat_{\mathcal{S}})^{\mathrm{c}})[W^{-1}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}((\mathcal{S})^{\mathrm{c}})[W^{-1}]\simeq{\mathcal{S}}_{\infty}.

This definition of $THH$ as a functor of $\infty$ -categories lets us deduce the following proposition from known properties of $THH$ in the setting of spectral categories.

0NQW

Proposition 10.2. $THH$ is a localizing invariant of small stable $\infty$ -categories.

0NQX

Proof. The cyclic bar construction commutes with filtered homotopy colimits of spectral categories. Furthermore, $THH(-)$ takes exact sequences of spectral categories to exact sequences of spectra [10, 7.1]. Therefore, the induced functor on $\infty$ -categories is a localizing invariant. ∎

The force of the co-representability result for algebraic $K$ -theory (theorem 7.13) is that it implies, via the spectral Yoneda lemma, the following identification of the spectrum of natural transformations of additive functors $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}E$ .

0NQY

Theorem 10.3. Given an additive invariant $E\colon\Cat_{\infty}^{\ex}\longrightarrow{\mathcal{S}}_{\infty}$ with values in the stable $\infty$ -category of spectra, we have a natural equivalence

\Nat(K,E)\simeq E({\mathcal{S}}_{\infty}^{\omega}),

where $\Nat(K,E)$ denotes the spectrum of natural transformations from $K$ to $E$ as additive invariants from small stable $\infty$ -categories to spectra.

0NQZ

Proof. By theorem 6.10, we can describe the additive invariants $K$ and $E$ as elements of $\mathrm{Fun}^{\mathrm{L}}({\mathcal{M}}_{\mathrm{add}},{\mathcal{S}}_{\infty})$ . The equivalence

\Nat(\mathrm{Map}({\mathcal{U}}_{\mathrm{add}}({\mathcal{S}}_{\infty}^{\omega}),-),E)\simeq E({\mathcal{S}}_{\infty}^{\omega})

follows from 7.13 and the spectral Yoneda lemma. ∎

In particular, applying theorem 10.3 to $THH$ yields the following corollary.

0NR0

Corollary 10.4. We have an equivalence of spectra

\Nat(K(-),THH(-))\cong THH({\mathcal{S}}^{\omega})\simeq THH(\mathbb{S})\simeq\mathbb{S}.

Passing to $\pi_{0}$ on both sides we obtain an isomorphism between homotopy classes of natural transformations and $\pi_{0}(\mathbb{S})\cong{\mathbb{Z}}$ .

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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10.1. T​H​HTHH as a localization invariant

10.1. $THH$ as a localization invariant