$K$-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**

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?

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?