 9:19 PM
Has anybody ever heard of a notion of a "seminorm" on a triangulated category $\mathcal C$? I'm thinking a map $|-|: Ob(\mathcal C) \to [0,\infty]$ such that $|0| = 0$ and $|B| \leq |A| + |C|$ for every triangle $A \to B \to C$. I'm toying with additionally requiring that there is some fixed $\rho > 0$ such that $|\Sigma A| = \rho |A|$ for all $A$, and maybe asking that $|A| \leq |A \oplus B|$ for all $A,B$. There are also some natural multiplicative properties to ask for...
The main example would be if you have a $t$-structure. Then you can define $|X| = \rho^n$ where $X \in \tau_{\geq n} \mathcal C \setminus \tau_{\geq n-1} \mathcal C$, at least under reasonable conditions.
The closest kind of thing that comes to mind is Bridgeland stability conditions, which are at least something assigning a number to each object of your category. But those are rather different.

1 hour later… 10:36 PM
@TimCampion This note of Amnon Neeman might be relevant: https://arxiv.org/abs/1901.01453 @AlexanderCampbell Nice, thanks! I will have to take a look!