ScalingStacks

Definition 8.1. Let ${\mathcal{D}}$ be a stable presentable $\infty$-category. A functor

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

is called a localizing invariant of small stable $\infty$-categories if it inverts Morita equivalences (see definition 2.14), preserves filtered colimits, and satisfies localization, i.e., sends exact sequences

$${\mathcal{A}}\longrightarrow{\mathcal{B}}\longrightarrow{\mathcal{C}}$$

of small stable $\infty$-categories (see definition 5.12) to cofiber sequences

$$E({\mathcal{A}})\longrightarrow E({\mathcal{B}})\longrightarrow E({\mathcal{C}})$$

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