ScalingStacks

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.