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Waldhausen introduced his categories for the purposes of defining algebraic $K$-theory of suitable categories. From a modern perspective, it looks like he was really doing two things at once:

  1. Waldhausen categories look like a fancier version of relative categories: a way to present a homotopy theory (i.e. $\infty$-category). That is, given a Waldhausen category $(C,cof,W)$, there ought to be some $\infty$-functor $$C \to Wald(C,cof,W)$$ to a pointed, finitely-cocomplete $\infty$-category which universally turns the morphisms of $W$ into equivalences and pushouts along morphisms of $cof$ into homotopy pushouts, and preserves the $0$ object.

NB: As I'm defining it, $Wald(C,cof,W)$ depends on both $cof$ and $W$. It ought to be constructed as follows. Let $Psh_\ast(C)$ be the category of pointed $Top_\ast$-valued presheaves on $C$ and $y: C \to Psh_\ast(C)$ the Yoneda embedding. Let $S = y(W) \cup \{ yb \cup_{ya} yc \to y(b \cup_a c) \mid b \leftarrowtail a \to c \}$, and let $L_S Psh_\ast(C)$ be the localization at these morphisms. Then $Wald(C,cof,W) \subseteq L_S(Psh_\ast(C))$ is the closure of the representables under finite colimits.

  1. Waldhausen $K$-theory appears to be an invariant associated to the $\infty$-category $Wald(C,cof,W)$, with a convenient presentation directly in terms of the data $(C,cof,W)$ via the $S_\bullet$ construction.

Now, I know of all sorts of ways to study the $\infty$-category presented by a category with weak equivalences $(C,W)$, depending on how nice $W$ is. And model categories give a way to study homotopy theory presented by similar data which includes cofibrations -- but it's well-known that in this case, the data of the cofibrations is redundant as far as the presented $\infty$-category goes, so that's a bit of a red herring. But what about when the Waldhausen category doesn't come from a model category?

Question 1: Has anybody studied the homotopy theory $Wald(C,cof,W)$ presented by a general Waldhausen category in the literature?

On the other hand, I know of several results in the literature showing that if one restricts to "nice" Waldhausen categories, the algebraic $K$-theory of the Waldhausen category is an invariant of its simplicial localization, and in particular the cofibrations are redundant for the purposes of $K$-theory. But in general, one would think that the cofibrations matter.

Question 2: Is it known in general whether Waldhausen $K$-theory is in fact an invariant of the $\infty$-category $Wald(C,cof,W)$ presented by a Waldhausen category $(C,cof,W)$?

Question 3: It's possible that I've been too naive in defining $Wald(C,cof,W)$. Should I perhaps consider something slightly different to be "the $\infty$-category presented by $(C,cof,W)$"?

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    $\begingroup$ Waldhausen categories are a ridiculously slick formalism solving a huge number of problems, like not having structured ring spectra. If you're trying to do $K$-theory, then, yeah, you should probably just work with ∞-categories. But Waldhausen was trying to relate $K$-theory to geometry. I think that there is a Waldhausen category where the equivalences are simple homotopy equivalences. The nerve of these equivalences is a space of interest. But it doesn't seem to be the maximal ∞-groupoid of an interesting (cocomplete) ∞-category. $\endgroup$
    – Ben Wieland
    Commented Sep 4, 2019 at 0:47
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    $\begingroup$ I think the review of modern $K$-theory given by Robalo in his thesis §7.1 is very well-done and might answer (positively) to most of your questions. $\endgroup$
    – Andrea Gagna
    Commented Sep 4, 2019 at 12:29
  • $\begingroup$ @BenWieland Good point. I'm confused about what lesson to draw. On the one hand, I might think that since simple homotopy equivalences don't satisfy 2/3, one shouldn't just localize at them. But then as you say, the space of simple homotopy equivalences is known to be interesting, so apparently it is interesting to localize at them... $\endgroup$
    – Tim Campion
    Commented Sep 8, 2019 at 21:08
  • $\begingroup$ Regarding Question 2, I'm now skeptical that the answer should be affirmative unless $(C,cof,W)$ has the property that every morphism is weakly equivalent to a cofibration, which is essentially the type of hypothesis used to show the cofibrations don't matter in the literature... After all, the $S_\bullet$ construction cares which morphisms are "cofibrations". $\endgroup$
    – Tim Campion
    Commented Sep 8, 2019 at 21:10
  • $\begingroup$ Simple homotopy equivalences do satisfy 2/3. But inverting them them inverts all homotopy equivalences see... A counterexample to 2 is that the category of finite sets is, technically, cocomplete as an $\infty$-category. If you take $S_\bullet$ for all morphisms you get zero, whereas $S_\bullet$ of monos is the sphere. But I'm not sure that this is really an interesting example; maybe one should just exclude it from the theory entirely. $\endgroup$
    – Ben Wieland
    Commented Sep 10, 2019 at 2:42

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