ScalingStacks

Definition 6.1. Let ${\mathcal{D}}$ be a stable presentable $\infty$-category. A functor

$$E:\Cat_{\infty}^{\ex}\longrightarrow{\mathcal{D}}$$

is called an additive invariant of small stable $\infty$-categories if it inverts Morita equivalences (see definition 2.14), preserves filtered colimits, and satisfies additivity, i.e., given a split-exact sequence

(6.2)

the functors $i$ and $g$ induce an equivalence in ${\mathcal{D}}$

$$E({\mathcal{A}})\vee E({\mathcal{B}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}E({\mathcal{C}})\,.$$

We denote by $\mathrm{Fun}_{\mathrm{add}}(\Cat_{\infty}^{\ex},{\mathcal{D}})$ the $\infty$-category of additive invariants with values in ${\mathcal{D}}$.