Proposition 9.31 (0NPY) · ScalingStacks

0NPY

Proposition 9.31. Let $\mathcal{C}$ be a presentable $\infty$ -category with a zero object, and let $\mathcal{C}^{\omega}[\Sigma^{-1}]$ denote the colimit

\mathcal{C}^{\omega}[\Sigma^{-1}]\simeq\colim\{\mathcal{C}^{\omega}\overset{\Sigma}{\longrightarrow}\mathcal{C}^{\omega}\overset{\Sigma}{\longrightarrow}\cdots\}

in $\Cat_{\infty}^{\mathrm{Rex}}$ . Then $\mathcal{C}^{\omega}$ is stable, and the induced functor

\mathcal{C}^{\omega}[\Sigma^{-1}]\longrightarrow\Stab(\mathcal{C})

identifies the idempotent-completion of $\mathcal{C}^{\omega}[\Sigma^{-1}]$ with $\Stab(\mathcal{C})^{\omega}$ .

0NPZ

Proof. Let ${\mathcal{D}}$ be an idempotent-complete stable $\infty$ -category. Then

	$\displaystyle\mathrm{Fun}^{\ex}(\mathcal{C}^{\omega}[\Sigma^{-1}],{\mathcal{D}})$	$\displaystyle\simeq\lim\mathrm{Fun}^{\ex}(\mathcal{C}^{\omega},{\mathcal{D}})\simeq\lim\mathrm{Fun}^{\mathrm{L}}_{\omega}(\mathcal{C},\Ind({\mathcal{D}}))$
		$\displaystyle\simeq\mathrm{Fun}^{\mathrm{L}}_{\omega}(\Stab(\mathcal{C}),\Ind({\mathcal{D}}))\simeq\mathrm{Fun}^{\mathrm{\ex}}(\Stab(\mathcal{C})^{\omega},{\mathcal{D}}).$

Since $\Stab(\mathcal{C})^{\omega}$ is necessarily idempotent-complete, we conclude that it is equivalent to the idempotent-completion of $\mathcal{C}^{\omega}[\Sigma^{-1}]$ . ∎