daniel gratzer on Nostr: Huh, so perhaps a simple argument shows that the algebraic K theory of stable ...

Huh, so perhaps a simple argument shows that the algebraic K theory of stable categories is (canonically) op-lax monoidal. Notably, this uses essentially no calculation besides a small amount of tinkering with split Verdier sequences.

It suffices to show that \(k : \mathsf{Cat}_{\mathsf{st}} \to \mathcal{S}\) is an algebra in the symmetric monoidal structure on \(\mathsf{Fun}(\mathsf{Cat}_{\mathsf{st}}, \mathcal{S})\) induced by Day convolution. A quick calculation shows that the unit of this monoidal structure is the core functor. One can calculate that the localization \(L\) reflecting to the subcategory of additive grouplike functors is monoidal and so there is an induced monoidal structure on additive goruplike functors whose unit is the reflection of core... which is precisely algebraic K theory! Now we find that \(k\) has a unique algebra structure in this localization, which then becomes an algebra structure in the larger SMC of all functors.

I think if one squints hard enough, this is what's going on in Blumberg-Gepner-Tabuada's second paper, but sort of buried beneath the language of their motives stuff.