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:

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

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  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

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  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.