Proposition 4.9 (0NKJ) · ScalingStacks

Proposition 4.9. Let $\mathcal{C}$ be a small stable $\infty$ -category. The fully-faithful inclusion

\Upsilon(\mathcal{C})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}_{\Delta}(\mathfrak{C}[\mathcal{C}]^{\op},{\mathcal{S}})

factors, on the level of underlying $\infty$ -categories, as the composite

	$\displaystyle\mathrm{N}(\Omega^{\infty}\Upsilon(\mathcal{C}))\simeq\mathcal{C}\rightarrow\mathrm{Fun}^{\ex}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})$	^ω
	$\displaystyle\subseteq\mathrm{Fun}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})$	$\displaystyle\simeq\mathrm{N}\mathrm{Fun}_{\Delta}(\mathfrak{C}[\mathcal{C}]^{\op},{\mathcal{S}})^{\cf}[W^{-1}].$

0NKK

Proof. Any stably representable functor $\mathcal{C}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ is exact, giving the factorization

\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})\subseteq\mathrm{Fun}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty}).

By proposition 3.2, we may rewrite this as $\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind(\mathcal{C})\simeq\mathrm{Fun}^{\ex}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})$ to see that, as an exact functor $\mathcal{C}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ , any stably representable functor is also compact. ∎