ScalingStacks

Theorem 1.1. (see theorems 6.10 and 8.7) There are stable presentable $\infty$-categories ${\mathcal{M}}_{\mathrm{add}}$ and ${\mathcal{M}}_{\mathrm{loc}}$ and universal additive and localizing invariants

(1.2)

$\displaystyle{\mathcal{U}}_{\mathrm{add}}\colon\Cat_{\infty}^{\ex}\longrightarrow{\mathcal{M}}_{\mathrm{add}}$

$\displaystyle{\mathcal{U}}_{\mathrm{loc}}\colon\Cat_{\infty}^{\ex}\longrightarrow{\mathcal{M}}_{\mathrm{loc}}\,.$

That is, given any stable presentable $\infty$-category ${\mathcal{D}}$, we have induced equivalences of $\infty$-categories

	$\displaystyle({\mathcal{U}}_{\mathrm{add}})^{*}\colon\mathrm{Fun}^{\mathrm{L}}({\mathcal{M}}_{\mathrm{add}},{\mathcal{D}})$	$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}$	$\displaystyle\mathrm{Fun}_{\mathrm{add}}(\Cat_{\infty}^{\ex},{\mathcal{D}})$
	$\displaystyle({\mathcal{U}}_{\mathrm{loc}})^{*}\colon\mathrm{Fun}^{\mathrm{L}}({\mathcal{M}}_{\mathrm{loc}},{\mathcal{D}})$	$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}$	$\displaystyle\mathrm{Fun}_{\mathrm{loc}}(\Cat_{\infty}^{\ex},{\mathcal{D}})\,,$

where the left-hand sides denote the $\infty$-categories of colimit-preserving functors and the right-hand sides the $\infty$-categories of additive and localizing invariants.