Waldhausen $K$-theory before group completion

Question

$K$-theory is often billed as the "universal way to split exact sequences". But it seems we're too anxious to group-complete things to actually take the slogan at face value.

Consider the following $\infty$-categories:

• $\mathcal{W}$ - Waldhausen categories (or Waldhausen $\infty$-categories, if you prefer)

• $\mathcal{C}_1$ - symmetric monoidal $\infty$-categories

• $\mathcal{C}_2$ - $E_\infty$-spaces

• $\mathcal{C}_3$ - infinite loop spaces

Say that a functor $F: \mathcal{W} \to \mathcal{C}_i$ is additive if $F\mathcal{E} W\to F W \times F W$ is an equivalence for all $W \in \mathcal{W}$, where $\mathcal{E}W$ the Waldhausen category of exact sequences $w' \to w \to w''$ in $W$ and the map projects onto $(w',w'')$. Consider the following functors:

• $K_\oplus^1: \mathcal{W} \to \mathcal{C}_1$ - sending $W$ to its simplicial localization, with $E_\infty$ structure given by coproduct
• $K_\oplus^2: \mathcal{W} \to \mathcal{C}_2$: - sending $W$ to the nerve of its category of weak equivalences (the core of $K_\oplus^1 W$)
• $K_\oplus^3 : \mathcal{W} \to \mathcal{C}_3$: - sending $W$ to the group completion of $K_\oplus^2 W$

The universal property of $K$-theory is that it is a functor $K^3: \mathcal{W} \to \mathcal{C}_3$ which constitutes a reflection of $K^3_\oplus$ into the category of additive functors $\mathcal{W} \to \mathcal{C}_3$ (I think this is Clark Barwick's formuation). Analogously, I ask:

Questions

1. Does $K^1_\oplus$ admit a reflection $K^1$ into the category of additive functors $\mathcal{W} \to \mathcal{C}_1$? If so, is it modeled by a variant of the $S_\bullet$ construction?

2. Does $K^2_\oplus$ admit a reflection $K^2$ into the category of additive functors $\mathcal{W} \to \mathcal{C}_2$? If so, is it modeled by the $S_\bullet$ construction itself?

Answer 1

I'm not sure about Waldhausen categories in general, but if you restrict attention to stable $\infty$-categories (with trivial Waldhausen structure in which all maps are cofibrations) then group completion is essentially forced on you by the splitting of exact sequences. In principle, this happens because for every object $X \in {\cal C}$, splitting the exact sequence $$ X \to 0 \to \Sigma X $$ means that $\Sigma X$ becomes the inverse of $X$. In particular, taking the maximal subgroupoid of a stable $\infty$-category ${\cal C}$ (considered as an $\mathbb{E}_\infty$-monoid in spaces with respect to direct sum), and then forcing the relations given by splitting of exact sequences, yields an $\mathbb{E}_\infty$-monoid which is already group-like.

This idea can also be phrased from the perspective of additivity. Observe first that the $\infty$-category ${\rm Cat}^{\rm ex}_\infty$ of small stable $\infty$-categories is semi-additive in the sense that it has a zero-object (an initial object which is also final) and biproducts (coproducts which are also products). Since the $\infty$-category of $\mathbb{E}_\infty$-monoids is also semi-additive it is somewhat natural to restrict attention to functors $F:{\rm Cat}^{\rm ex}_\infty \to {\rm Mon_{\mathbb{E}_\infty}}$ which are themselves semi-additive, in the sense that they preserve zero-objects and biproducts. Note that so far we have not enforced any group-likeness condition. Note also that such a semi-additive $F$ will be additive in the sense you describe if and only if it sends the map $p:{\cal EC} \to {\cal C}\times{\cal C}$ given by $[x' \to x \to x''] \mapsto (x',x'')$ to an equivalence of $\mathbb{E}_\infty$-monoids. We now claim the following: if a semi-additive functor $F:{\rm Cat}^{\rm ex}_\infty \to {\rm Mon_{\mathbb{E}_\infty}}$ is additive then it automatically takes values in group-like $\mathbb{E}_\infty$-spaces. To see this observe that when ${\cal C}$ is stable the $\infty$-category ${\cal EC}$ of fiber sequences has an automorphism $T: {\cal EC} \to {\cal EC}$ sending $[x'\to x \to x'']$ to $[x \to x'' \to {\rm cof}(x\to x'')]$. Furthermore, the functor $p:{\cal EC} \to {\cal C}\times{\cal C}$ above has a one-sided inverse $i: {\cal C} \times {\cal C} \to {\cal EC}$ sending $(x,y)$ to $[x \to x\oplus y \to y]$. In particular, if $F(p)$ is an equivalence then $F(i)$ is an equivalence as well. It then follows that if $F$ is additive then it sends the composed functor $$ {\cal C} \times {\cal C} \stackrel{i}{\to} {\cal EC} \stackrel{T}{\to} {\cal EC} \stackrel{p}{\to} {\cal C} \times {\cal C} \stackrel{\sigma}{\to} {\cal C} \times {\cal C} $$ to an equivalence, where $\sigma$ is the equivalence $\sigma(x,y) = (\Omega y,x)$. But this composed functor is just the shear map $(x,y) \mapsto (x,x\oplus y)$ and since $F$ is semi-additive it follows that the shear map of the $\mathbb{E}_\infty$-monoid $F({\cal C})$ is an equivalence, i.e., $F({\cal C})$ is group-like.