I am studying K-theory with Weibel's K-book and have just read the definition of the $S.$ construction for Waldhausen categories. I also recently watched a series of talks by Thomas Nikolaus which discusses a more abstract framework to formulate K-theory. I don't yet have the infinity categorical baggage to understand all those ideas. However they did inspire my question.
Is there a way to formulate $S.$ construction not as an explicit (currently ad hoc) simplicial set, but instead as an adjoint. A (to me) natural guess would be that the $S.$ construction might fit into an adjunction with the functor from simplicial Waldhausen categories to Waldhausen categories sending $\mathcal{C}_\bullet$ to $\mathcal{C}_1$.
Thank you to anyone who finds the time to think about this.
As I wrote in a comment, the adjunction that you propose does hold in an appropriate $\infty$-categorical version of the $S$-construction. I will quickly explain what stable $\infty$-categories are (and how they are related to Waldhausen categories), and how we obtain the functor $S\colon\mathrm{Cat}^\mathrm{st}_\infty\to\mathrm{sCat}^\mathrm{st}_\infty$, where $\mathrm{Cat}^\mathrm{st}_\infty$ is the $\infty$-category of stable $\infty$-categories, and $\mathrm{sCat}^\mathrm{st}_\infty$ is the $\infty$-category of simplicial stable $\infty$-categories, i.e. functors $\Delta^\mathrm{op}\to\mathrm{Cat}^\mathrm{st}_\infty$. After that, I will sketch the argument why $S$ is a left adjoint. A complete proof is a bit longer, and I think the level of formality required for it does not help to understand the idea of the argument any better.
Stable $\infty$-categories
As you may know, an $\infty$-category is a joint generalization of ordinary categories and homotopy types, and a very good place to do abstract homotopy theory. Its higher morphisms serve as abstractions of higher homotopies between its $1$-morphisms (and the $n$-morphisms for $n\geq 2$ are consequently all required to be invertible in a suitable sense). Every time I talk about things like pushouts, pullbacks, universal properties and the like, you should think about these as homotopy pushouts, homotopy pullbacks, and ''homotopical'' universal properties. In a Waldhausen category, only the pushout squares featuring a cofibration model the idea of homotopy pushouts correctly, but in the $\infty$-categorical world any pushout does that. Among other things, this makes the construction of the $S$-construction easier.
It does also mean that we do not necessarily want to look at Waldhausen categories anymore as source of the $S$-construction. There are a couple types of $\infty$-categories that play a role similar to the homotopical concept that Waldhausen categories are supposed to model. I will focus on stable $\infty$-categories. These can also be seen as the ''correct'' enhancement of the notion of a triangulated category to the $\infty$-categorical world. (For instance, the homotopy $1$-category of any stable $\infty$-category carries a canonical triangulated category structure, and any triangulated category that you actually encounter in nature can be realized as such. The pathetic counterexamples are for me just proof that the definition of a triangulated category is not perfect.)
Definition. A stable $\infty$-category is an $\infty$-category with a zero object, finite limits and finite colimits, such that a commutative square is a pushout square iff it is a pullback square.
For example, the derived $\infty$-category of any ring or additive category is stable, and the $\infty$-category of spectra is as well. From now on, let $\mathscr{C}$ be a stable $\infty$-category.
Fact. Finite coproducts in $\mathscr{C}$ coincide with finite products, i.e. $\mathscr{C}$ has biproducts.
Definition. Given an object $x$ in $\mathscr{C}$, its suspension is the object $\Sigma x$ given as pushout $$\require{AMScd}\begin{CD} x @>>> 0 \\ @VVV @VVV\\ 0 @>>> \Sigma x\end{CD}$$ and its loop object is the object $\Omega x$ given as pullback $$\require{AMScd}\begin{CD} \Omega x @>>> 0\\ @VVV @VVV\\ 0@>>> x\end{CD}$$ You can show (because pushout squares and pullback squares coincide in $\mathscr{C}$ that the functors $\Sigma\colon\mathscr{C}\rightleftarrows\mathscr{C}\colon\Omega$ that we obtain in this way are inverse equivalences of $\infty$-categories. This is why we call such $\infty$-categories stable.
Definition. Given a morphism $f\colon x\to y$ in $\mathscr{C}$, its cofiber is the object $y/x$ (or rather, the morphism $y\to y/x$) given by the pushout square $$\require{AMScd}\begin{CD} x@>{f}>> y\\ @VVV @VVV\\ 0 @>>> y/x\end{CD}$$ and its fiber is the object $\mathrm{fib}(f)$ given by the pullback square $$\require{AMScd}\begin{CD} \mathrm{fib}(f) @>>> x\\ @VVV @VV{f}V\\ 0 @>>> y\end{CD}$$ The cofiber is also usually denoted by $\mathrm{cofib}(f)$. You can show that $\Sigma\mathrm{fib}(f)\simeq y/x$. This recovers all kinds of long exact sequences in algebraic topology.
In general, a stable $\infty$-category is a place you can do some form of stable homotopy theory or homological algebra in, albeit with the understanding that all fibers and cofibers are homotopy fibers and homotopy cofibers (if you come from a more algebraic background, these are the derived fibers and derived quotients you may have seen in the derived category of a ring). Its axioms are slightly stronger than the analogous axioms of Waldhausen categories (for instance, we assume finite limits exist), but we do not need any notion of cofibrations because our pushouts have automatically the correct homotopical behaviour.
Definition. An exact functor $F\colon\mathscr{C}\to\mathscr{D}$ between stable $\infty$-categories is a functor that preserves finite limits and finite colimits. In particular, it preserves zero objects, biproducts, suspensions, loop objects, fibers and cofibers. We write $\mathrm{Cat}^\mathrm{st}_\infty$ for the $\infty$-category of stable $\infty$-categories and exact functors (and the usual higher morphisms between functors).
Observation. A functor $F$ like above is exact already if it just preserves finite colimits: pullbacks are preserved since these squares are pushout squares as well, and finite products are preserved since these are the same as finite coproducts.
The $S$-construction
Given our stable $\infty$-category $\mathscr{C}$, we associate to it another stable $\infty$-category $S_n(\mathscr{C})$, for any $n\geq 0$. We let $[n]$ denote the ($1$-)category $0\to1\to\ldots\to n$, and write $\mathrm{Ar}([n])$ for the arrow category of this category, i.e. the functor category $\mathrm{Fun}([1],[n])$. You can think of it as the poset of pairs $(i,j)$ with $0\leq i\leq j\leq n$, ordered lexicographically. We let $S_n(\mathscr{C})$ be the full sub-$\infty$-category of the functor $\infty$-category $\mathrm{Fun}(\mathrm{Ar}([n]),\mathscr{C})$ on those functors $F$ for which the following conditions are satisfied:
This means that $S_0(\mathscr{C})$ is just the terminal $\infty$-category $*$. We picture $S_1(\mathscr{C})$ as the $\infty$-category of diagrams
and although this is just equivalent to $\mathscr{C}$ itself, this presentation is extra useful if we picture $S_2(\mathscr{C})$ as the $\infty$-category of diagrams
For ease of notation, we write an object of $S_1(\mathscr{C})$ as $0\to x\to 0$, and an object of $S_2(\mathscr{C})$ as $0\to x\to y\to y/x\to 0$. (Notice the similarity in notation with a short exact sequence.) In general, $S_n(\mathscr{C})$ is obtained by taking a sequence $0\to x_1\to x_2\to\ldots\to x_n$ of maps in $\mathscr{C}$, and iteratively taking cofibers and pushouts of these maps as to obtain a diagram of the shape
in $\mathscr{C}$. $S_n(\mathscr{C})$ is the $\infty$-category of all such diagrams in $\mathscr{C}$.
Fact. Each $S_n(\mathscr{C})$ is a stable $\infty$-category (in fact, there is an equivalence $S_n(\mathscr{C})\simeq\mathrm{Fun}([n-1],\mathscr{C})$ of $\infty$-categories). Its corresponding suspension functor is defined on diagrams by objectwise applying the suspension functor $\Sigma\colon\mathscr{C}\to\mathscr{C}$.
The collection $\{[n]\mid n\geq 0\}$ carries a cosimplicial structure, and hence we can assemble the collection $S_n(\mathscr{C})$ for $n\geq 0$ into a single simplicial object $S(\mathscr{C})$ in $\mathrm{Cat}^\mathrm{st}_\infty$. This is the reason why we did not just take $S_n(\mathscr{C})$ to be defined by $\mathrm{Fun}([n-1],\mathscr{C})$, because the latter does not have obvious ''degree $n$'' simplicial structure.
For instance, the degeneracy operations $S_2(\mathscr{C})\to S_1(\mathscr{C})$ are as follows: an object $0\to x\to y\to y/x\to 0$ in $S_2(\mathscr{C})$ is sent by $d_0$ to the object $0\to y/x\to 0$, by $d^1$ to $0\to y\to 0$, and by $d_2$ to $0\to x\to 0$. And an object $0\to x\to 0$ in $S_1(\mathscr{C})$ is sent by the degeneracies $s_0,s_1\colon S_1(\mathscr{C})\to S_2(\mathscr{C})$ to the objects $0\to x\xrightarrow{\mathrm{id}_x} x\to 0\to 0$ and $0\to 0\to x\xrightarrow{\mathrm{id}_x} x \to 0$ respectively.
Any exact functor $F\colon\mathscr{C}\to\mathscr{D}$ between stable $\infty$-categories induces an exact functor $S_n(\mathscr{C})\to S_n(\mathscr{D})$ for all $n\geq 0$ by postcomposition, and we obtain a functor $S\colon\mathrm{Cat}^\mathrm{st}_\infty\to\mathrm{sCat}^\mathrm{st}_\infty$, where the target is the $\infty$-category of simplicial objects in $\mathrm{Cat}^\mathrm{st}_\infty$, i.e. the functor category $\mathrm{Fun}(\Delta^\mathrm{op},\mathrm{Cat}^\mathrm{st}_\infty)$.
Definition. The Waldhausen $S$-construction is this functor $S\colon\mathrm{Cat}^\mathrm{st}_\infty\to\mathrm{sCat}^\mathrm{st}_\infty$.
Fun fact. Waldhausen himself called this the Segal $S$-construction, and the $S$ is sometimes said to stand for ''Segal''. Some people will try to convince you that we should go back to calling it the Segal $S$-construction.
The adjunction
Write $\mathrm{ev}_1\colon\mathrm{sCat}^\mathrm{st}_\infty\to\mathrm{Cat}^\mathrm{st}_\infty,\mathscr{C}_\bullet\to\mathscr{C}_1$ for the evaluation functor.
Proposition. There is an adjunction $S\colon\mathrm{Cat}^\mathrm{st}_\infty\rightleftarrows\mathrm{sCat}^\mathrm{st}_\infty\colon\mathrm{ev}_1$ of $\infty$-functors.
Sketch of the proof. We will make it believable that there is an equivalence $\mathrm{sCat}^\mathrm{st}_\infty(S(\mathscr{C}),\mathscr{D}_\bullet)\simeq\mathrm{Cat}^\mathrm{st}_\infty(\mathscr{C},\mathscr{D}_1)$ of spaces, for any stable $\infty$-category $\mathscr{C}$ and any simplicial stable $\infty$-category $\mathscr{D}_\bullet$. Consider namely a functor $F_\bullet\colon S(\mathscr{C})\to\mathscr{D}_\bullet$ of simplicial stable $\infty$-categories. Since $S_1(\mathscr{C})\simeq \mathscr{C}$, $F_1$ can be seen as an exact functor $\mathscr{C}\to\mathscr{D}_1$. Hence we must make it believable that $F$ is uniquely determined (up to contractible ambiguity) by $F_1$. We will show that $F_2$ can be recovered from $F_1$ alone. A similar argument will show that $F_n$ can be too for all $n\geq 2$.
Given an object $0\to x\to y\to y/x\to 0$ of $S_2(\mathscr{C})$, we can find an object $0\to w\to z\to z/w\to 0$ such that the sequence $0\to z\to z/w\to (z/w)/z\to 0$ is equivalent to $0\to x\to y\to y/x\to 0$. In fact, we can take $w\simeq\mathrm{fib}(x\to y)$ and $z\simeq x$. That this works is a nice exercise about the basic definitions of concepts in stable $\infty$-categories we made above. Now, another nice exercise about those is to now prove that the object $0\to x\to y\to y/x\to 0$ is actually the cofiber in $S_2(\mathscr{C})$ of the map $$ (0\to 0\to w\xrightarrow{\mathrm{id}_w} w\to 0)\to(0\to z\xrightarrow{\mathrm{id}_z} z\to 0\to 0). $$ This map is a map $s_1(0\to w\to 0)\to s_0(0\to z\to 0)$, with $s_0,s_1\colon S_1(\mathscr{C})\to S_2(\mathscr{C})$ the degeneracy operators we described earlier. But $F_2$ needs to preserve cofibers and needs to respect degeneracies. Therefore $F_2(0\to x\to y\to y/x\to 0)$ is the cofiber in $\mathscr{D}_2$ of a map $s_1F_1(0\to w\to 0)\to s_0F_1(0\to z\to 0)$. We only still need to check that the particular map used here is also determined by $F_1$ in some way. However, the only nontrivial part of the map $s_1(0\to w\to 0)\to s_0(0\to z\to 0)$ (i.e., where we do not map from a zero object, or to a zero object in our diagram; notice that zero objects in our diagrams are preserved by $F_2$ because $F_2$ preserves face operators and $F_0\colon *\to \mathscr{D}_0$ is an exact functor) is the map $w\to z$ appearing at position $(2,2)$. That means that the only part of the map that is not uniquely determined (up to contractible choice, as usual) a priori is the induced map $d_1s_1(0\to w\to 0)\to d_1s_0(0\to z\to 0)$ in $S_1(\mathscr{C})$. But since $F_2$ needs to preserve face operators, it is already determined how it must act on this map in terms of $F_1$. All in all, the functor $F_2$ is fully determined by $F_1$. As said, a similar argument will prove that $F_n$ is fully determined by $F_1$ for $n\geq 2$. The functor $F_0\colon *\to \mathscr{D}_0$ is uniquely determined by it being exact as well, so $F_\bullet\colon S(\mathscr{C})\to\mathscr{D}_\bullet$ is uniquely determined by $F_1$.
If you want to make this proof precise, you would use the recipe described in it to define a counit transformation $\varepsilon_\bullet\colon S(\mathscr{D}_1)\to\mathscr{D}_\bullet$ for which $\varepsilon_1\colon S_1(\mathscr{D}_1)\to\mathscr{D}_1$ is the equivalence we also used above. The unit transformation is simply this same equivalence $\mathscr{C}\simeq S_1(\mathscr{C})$ (or, to be pedantic, an inverse to it). Then one could verify that the triangle identities are satisfied in a suitable sense, which essentially boils down to the above argument that $F_\bullet$ was fully determined by $F_1$.
I want to stress that although all of this takes place in $\infty$-categories, none of this should be considered scary. In fact, the sketch of the proof of the adjunction is actually quite simple, as were the definitions of all the concepts! This is the power of $\infty$-categories: complicated homotopical structures are turned into concrete objects that we can manipulate a bit as if we were working with ordinary categories. Let us end with a nice corollary of the proof.
Corollary. The functor $S\colon\mathrm{Cat}^\mathrm{st}_\infty\to\mathrm{sCat}^\mathrm{st}_\infty$ is fully faithful.
Proof. We just saw that the unit of the adjunction $S\colon\mathrm{Cat}^\mathrm{st}_\infty\rightleftarrows\mathrm{sCat}^\mathrm{st}_\infty\colon\mathrm{ev}_1$ is an equivalence on all objects. Just like in ordinary category theory, this forces $S$ to be fully faithful.
What about the case of Waldhausen categories?
The $\infty$-categorical $S$-construction as defined above does not need that our $\infty$-category $\mathscr{C}$ has finite limits at all, just that it has a zero object. However, in the second paragraph of the proof of the adjunction I have used fibers of maps in an essential way, so this proof does need that we work with stable $\infty$-categories. I do not know if you can generalize the result to general $\infty$-categories with a zero object and finite colimits; I do not have a counterexample to show we cannot. This is one reason why the adjunction may fail for ordinary Waldhausen categories, as something like finitely cocomplete $\infty$-categories with a zero object may be the more proper analogue of Waldhausen categories than stable $\infty$-categories. Another reason is the usage of cofibrations in the original $S$-construction, and the fact that all of that is just a lower-categorical model for something that wants to be an $\infty$-functor. In particular, it stands to reason that the adjunction only wants to be an $\infty$-adjunction, and $\infty$-adjunctions are not always modeled by lower-categorical adjunctions of presentations of the constructions.