ScalingStacks

Waldhausen’s algebraic $K$-theory functor takes as input a category with cofibrations and weak equivalences. It is now well understood that, under mild hypotheses, the $K$-theory spectrum is determined by the Dwyer-Kan localization $L^{H}{\mathcal{C}}$ of the Waldhausen category ${\mathcal{C}}$ [81, 13, 19]. Since $\mathrm{N}((L^{H}{\mathcal{C}})^{\textrm{fib}})$ yields the $\infty$-category associated to ${\mathcal{C}}$, these results can be interpreted as saying that the algebraic $K$-theory of a Waldhausen category is an invariant of the underlying $\infty$-category. Moreover, it has long been folklore that given a sufficiently good theory of $\infty$-categories one can define analogues of Waldhausen’s construction of algebraic $K$-theory (e.g., see [81, §7] for a sketch of such a definition in the context of Segal categories). In this subsection we study a version of such a direct construction of the algebraic $K$-theory of $\infty$-categories in the setting of quasicategories [53, 1.2.2.5]. We prove that Waldhausen’s algebraic $K$-theory of a Waldhausen category ${\mathcal{C}}$ is equivalent as a spectrum to this $\infty$-categorical algebraic $K$-theory of the associated $\infty$-category $\mathrm{N}((L^{H}{\mathcal{C}})^{\mathrm{fib}})$.

We begin by reviewing Waldhausen’s $S_{\bullet}$ construction. Let ${\mathcal{C}}$ be a Waldhausen category. Let $\Ar[n]$ denote the category of arrows in $[n]$: $\Ar[n]$ has objects $(i,j)$ for $0\leq i\leq j\leq n$ and a unique map $(i,j)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(i^{\prime},j^{\prime})$ for $i\leq i^{\prime}$ and $j\leq j^{\prime}$. Then $S_{n}{\mathcal{C}}$ is the full subcategory of the category of functors $A\colon\Ar[n]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ such that:

• •

  $A_{i,i}=*$ for all $i$,

• •

  The map $A_{i,j}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{i,k}$ is a w-cofibration for all $i\leq j\leq k$, and

• •

  The diagram

  is a pushout square for all $i\leq j\leq k$,

The algebraic $K$-theory space of ${\mathcal{C}}$ is then defined to be $\Omega|w_{\bullet}S_{\bullet}{\mathcal{C}}|$, where the weak equivalences in $S_{\bullet}{\mathcal{C}}$ are defined pointwise. Furthermore, since each $S_{n}{\mathcal{C}}$ is itself a Waldhausen category (with the Reedy cofibrations), we can iterate the $S_{\bullet}$ construction. The algebraic $K$-theory spectrum of ${\mathcal{C}}$ is the spectrum with $n$th space $|w_{\bullet}S_{\bullet}^{(n)}{\mathcal{C}}|$.

Now let ${\mathcal{C}}$ be a small pointed $\infty$-category with finite colimits. The following definition [53, 1.2.2.2] is the $\infty$-categorical analogue of Waldhausen’s $S_{\bullet}$ construction.

As with the classical $S_{\bullet}$ construction, when ${\mathcal{C}}$ has all colimits, the data of the cocartesian squares (i.e., cofibers for the maps $F(i,j)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}F(i,k)$) is necessary only for the simplicial structure.

Following [53, 1.2.2.5], we define a simplicial $\infty$-category $S^{\infty}_{\bullet}{\mathcal{C}}$ by the rule $S^{\infty}_{n}\mathcal{C}=\Gap([n],{\mathcal{C}})$. Applying passage to the largest Kan complex levelwise, we obtain a simplicial space $(S^{\infty}_{\bullet}{\mathcal{C}})_{\mathrm{iso}}$. Then $\Omega|(S^{\infty}_{\bullet}{\mathcal{C}})_{\mathrm{iso}}|$ is the $\infty$-categorical version of Waldhausen’s $K$-theory space. Furthermore, for each $n$, $\Gap([n],{\mathcal{C}})$ is itself a small pointed $\infty$-category with finite colimits: once again, we can can iterate this procedure. Since $\Gap([0],{\mathcal{C}})$ is contractible (with preferred basepoint given by the point in ${\mathcal{C}}$) and $\Gap([1],{\mathcal{C}})$ is equivalent to ${\mathcal{C}}$, there is a natural map

given by the inclusion into the $1$-skeleton. Therefore, the spaces $|((S^{\infty}_{\bullet})^{n}({\mathcal{C}}))_{\mathrm{iso}}|$ assemble to form a spectrum $K({\mathcal{C}})$; this is the $\infty$-categorical version of Waldhausen’s $K$-theory spectrum. We can see from the definition that $\Gap([n],{\mathcal{C}})$ is natural in (right) exact functors, and therefore $S^{\infty}_{\bullet}{\mathcal{C}}$ and $|(S^{\infty}_{\bullet}{\mathcal{C}})_{\mathrm{iso}}|$ are also natural. Since the equivalence $\Gap([1],{\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}(*,{\mathcal{C}})\simeq{\mathcal{C}}$ induced by the restriction map is natural in ${\mathcal{C}}$, we deduce that the $K$-theory spectrum is natural in exact functors.

In practice, we find it more convenient to use an “all at once” reformulation of the definition of the iterated $S_{\bullet}$ construction (e.g., see [13, A.5.4], [14, 2.2], the appendix to [34], and also [67, §2]).

Now we define the multisimplicial $\infty$-category

We regard $(S^{\infty}_{\bullet})^{(0)}$ as ${\mathcal{C}}$ and it is clear that $(S^{\infty}_{\bullet})^{(1)}_{n}$ is $\Gap([n],{\mathcal{C}})$. Now we directly the define the $K$-theory spectrum of an $\infty$-category ${\mathcal{C}}$ with finite colimits to be the spectrum with $q$-th space

The suspension maps $\Sigma K{\mathcal{C}}(q)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}K(q+1)$ are induced on diagrams by the projection map

From definition 7.5 it is now clear that the construction of the $K$-theory spectrum is functorial in (right) exact functors.

We now establish a comparison between Waldhausen’s algebraic $K$-theory of a Waldhausen category ${\mathcal{C}}$ and the $\infty$-categorical version of the algebraic $K$-theory of the associated simplicial category $L^{H}{\mathcal{C}}$. The comparison is essentially a consequence of the theory of rigidification of homotopy coherent diagrams to strict diagrams in a model category (originally studied by Dwyer-Kan [30]), which allows us to pass between $\infty$-categorical diagrams and point-set diagrams, and the “homotopical” $S^{\prime}_{\bullet}$ construction of [12], which allows us to replace the use of pushouts by homotopy pushouts for suitable Waldhausen categories. The version of the comparison of homotopy coherent diagrams to strict diagrams we use is originally due to Hirschowitz-Simpson [43] in the context of Segal categories (see also Rezk’s work in Segal spaces [77, 8.12]). Since we are using quasicategories in this paper, we work with the version proved by Lurie in that setting [52, 4.2.4.4].

Let $S$ be a small simplicial set, ${\mathcal{D}}$ a small simplicial category, and $u\colon\mathfrak{C}[S]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ an equivalence. Let ${\mathcal{A}}$ be a combinatorial simplicial model category, and let ${\mathcal{U}}$ be a ${\mathcal{D}}$-chunk of ${\mathcal{A}}$ (see [52, A.3.4.9] for a discussion of ${\mathcal{D}}$-chunks). Then the induced map

is a categorical equivalence of simplicial sets. Here the notation $({\mathcal{U}}^{{\mathcal{D}}})^{\cf}$ indicates the full subcategory of ${\mathcal{A}}^{{\mathcal{D}}}$ consisting of cofibrant-fibrant objects (in the projective model structure) landing in ${\mathcal{U}}$.

Specializing to our situation, assume that $S$ is the (ordinary) nerve $\mathrm{N}(J)$ of a diagram (small category) $J$; that is, $J$ is regarded as a discrete simplicial category. Then the counit map $\mathfrak{C}[\mathrm{N}(J)]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}J$ is an equivalence and so we have that the induced map

is a categorical equivalence of simplicial sets.

To apply these rigidification results, we use the $S^{\prime}_{\bullet}$ construction. The $S^{\prime}_{\bullet}$ construction [12, 2.7] is a variant of Waldhausen’s $S_{\bullet}$ construction defined by replacing the cocartesian squares in the definition of $S_{\bullet}$ with homotopy cocartesian squares. In order to define the $S^{\prime}_{\bullet}$ construction, we must work with Waldhausen categories for which there is a reasonable notion of homotopy cocartesian squares. We briefly recall this theory from [12, §2]. A map is a weak cofibration if it is equivalent by a zig-zag to a cofibration, and a square is a homotopy cocartesian square if it equivalent by a zig-zag to a pushout square with one leg a cofibration. For control on these notions, we require the hypothesis that any map in ${\mathcal{C}}$ can be factored (not necessarily functorially) as a cofibration followed by a weak equivalence.

For such a Waldhausen category ${\mathcal{C}}$, we can then define $S^{\prime}_{n}{\mathcal{C}}$ to be the full subcategory of the category of functors $\Ar[n]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ such that:

• •

  $A_{i,i}\simeq*$ for all $i$,

• •

  The map $A_{i,j}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A_{i,k}$ is a weak cofibration for all $i\leq j\leq k$, and

• •

  The diagram

  is a homotopy cocartesian square for all $i\leq j\leq k$,

By construction, the $S^{\prime}_{\bullet}$ construction is functorial in weakly exact functors, i.e., functors that preserve weak equivalences and homotopy cocartesian squares. Moreover, the natural inclusion

is a weak equivalence [12, 2.9]. Therefore, we can equivalently define the algebraic $K$-theory space of a Waldhausen category ${\mathcal{C}}$ as $\Omega|w_{\bullet}S^{\prime}_{\bullet}{\mathcal{C}}|$ and similarly the algebraic $K$-theory spectrum of ${\mathcal{C}}$ as having $n$th space $|w_{\bullet}(S^{\prime}_{\bullet})^{(n)}{\mathcal{C}}|$.

Now, let ${\mathcal{C}}$ be a Waldhausen category that arises as a subcategory of a model category. Since a square is homotopy cocartesian in ${\mathcal{C}}$ if and only if it is a pushout square in $\mathrm{N}(({\mathcal{C}})^{\cf})$, in this setting the equivalence of Lemma 7.6 restricts to give an equivalence

Similar considerations for the iterated $S^{\prime}_{\bullet}$ construction [13, A.5.4] (as modeled in Definition 7.5) yield the equivalence

Applying proposition 2.10, we then obtain the following comparison of algebraic $K$-theory spaces and spectra.

In particular, this yields the following theorem:

Finally, specializing to our case, we find the following result.

As a consequence, Waldhausen’s additivity theorem applies to prove the following proposition.

So far, all of our comparison results assume that the Waldhausen category we are working with arises as a subcategory of a model category. In fact, we can extend our comparison and functoriality results to Waldhausen categories ${\mathcal{C}}$ such that all maps admit (not necessarily functorial) factorizations as cofibrations followed by weak equivalences and which are DKHS-saturated (i.e., such that a map $f$ is a weak equivalence in ${\mathcal{C}}$ if and only if its image in the homotopy category is an isomorphism). We do this as follows, using a construction due to Cisinski [19, §4].

As a corollary, we have the following result comparing of the $K$-theory of Waldhausen categories that are DHKS-saturated and admit factorization to the associated $K$-theory of $\infty$-categories.

Given a (homotopically) pointed simplicial category with finite homotopy colimits, we can use essentially the same construction to produce a DK-equivalent Waldhausen category; see [81, §5] and [10, §14] for versions of such a construction.

This subsection is entirely devoted to the proof of theorem 7.13. The proof will follow from propositions 7.17 and 7.19.

In the following results, we will use the observation that Waldhausen’s $S_{\bullet}$ construction, applied to a spectral category which is a Waldhausen category with the cofibrations inherited from a spectral model structure with all objects fibrant, produces a spectral category (where the mapping spectra are given by an appropriate end) [11, §3]. To ensure we are in this setting, we will tacitly use the equivalent model of spectral categories enriched in EKMM $S$-modules, as explained in [10, §15]. Alternatively, we could stay with spectral categories in symmetric spectra and use the “Moore” $S_{\bullet}$ construction from [11, §4], which uses an explicit model of the homotopy end. We also need the following lemma which allows us to bring the $S_{\bullet}$ construction inside:

We can now relate ${\mathcal{M}}_{\mathrm{add}}$ to the algebraic $K$-theory presheaf.

Propositions 7.17 and 7.19 allow us to prove theorem 7.13 as follows : let ${\mathcal{A}}$ be stable $\infty$-category and ${\mathcal{B}}$ a compact small idempotent-complete stable $\infty$-category. By proposition 7.17 we have an equivalence ${\mathcal{U}}_{\mathrm{add}}({\mathcal{A}})\simeq K_{{\mathcal{A}}}$ and by proposition 7.19 $K_{{\mathcal{A}}}$ is local. Therefore, we have the following natural equivalence

where the right-hand side is calculated in $\mathrm{Pre}((\Cat_{\infty}^{\perf})^{\omega};{\mathcal{S}}_{\infty})$. Since ${\mathcal{B}}$ belongs to $(\Cat_{\infty}^{\perf})^{\omega}$, the presheaf $\psi({\mathcal{B}})$ is representable and so by the spectral Yoneda lemma we have $\mathrm{Map}(\psi({\mathcal{B}}),K_{{\mathcal{A}}})\simeq K_{{\mathcal{A}}}({\mathcal{B}})$. Finally, since by definition of $K_{{\mathcal{A}}}$ we have $K_{{\mathcal{A}}}({\mathcal{B}})=K(\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}})))$ the proof is finished.