Proposition 4.15 (0NKU) · ScalingStacks

0NKU

Proposition 4.15. The functor $\eta_{{\mathcal{A}}}\colon{{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M{\mathcal{A}}$ is essentially surjective if and only if ${{\mathcal{A}}}$ is stable.

0NKV

Proof. Indeed, ${{\mathcal{A}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M{{\mathcal{A}}}$ is essentially surjective if and only if $\Omega^{\infty}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{\infty}M{\mathcal{A}}$ is essentially surjective, which is the case if and only if ${\mathcal{A}}$ is already stable. ∎