Theorem 6.10 (0NN0) · ScalingStacks

0NN0

Theorem 6.10. The functor ${\mathcal{U}}_{\mathrm{add}}$ is the universal additive invariant, i.e., given any stable presentable $\infty$ -category ${\mathcal{D}}$ , we have an equivalence of $\infty$ -categories

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

0NN1

Proof. The result follows from theorem 6.7 and from the universal property of stabilization (i.e., [53, 1.4.5.5]). Note that stabilization preserves colimits and ${\mathcal{U}}_{\mathrm{add}}^{\mathrm{un}}$ sends split-exact sequences to cofiber sequences, so the split-exact sequence (6.2) is sent to a split cofiber sequence ${\mathcal{U}}_{\mathrm{add}}({\mathcal{C}})\simeq{\mathcal{U}}_{\mathrm{add}}({\mathcal{A}})\vee{\mathcal{U}}_{\mathrm{add}}({\mathcal{B}})$ in ${\mathcal{M}}_{\mathrm{add}}$ . ∎