daniel gratzer on Nostr: Let me answer in reverse order: by stable category I do indeed mean a stable ...

Let me answer in reverse order: by stable category I do indeed mean a stable ∞-category. So in the above, I mean "the" algebraic K theory to be a functor from the category of stable categories to spaces which is additive and grouplike (preserves the terminal object and split Verdier pullback squares). One can give explicit descriptions of this using Quillen's Q construction or Wauldhausen's S construction, but we can also use the universal property of Blumberg-Gepner-Tabuada directly: it's the initial additive and group-like functor receiving a natural transformation from the core functor.

I learned about this stuff from Fabian Hebestreit's lectures as written up by Wagner (https://florianadler.github.io/AlgebraBonn/KTheory.pdf)