Proposition 2.17 (0NJQ) · ScalingStacks

0NJQ

Proposition 2.17. Let $\mathcal{C}$ be a stable $\infty$ -category. Then a functor $X\colon\mathcal{C}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ is stably representable if and only if it is represented by the suspension spectrum $\Sigma^{\infty}z$ of a unique (up to equivalence) object $z$ of $\mathcal{C}$ .

0NJR

Proof. It suffices to show that any spectrum object $A$ of $\mathcal{C}$ is of the form $\Sigma^{\infty}z$ for a uniquely determined object $z$ of $\mathcal{C}$ . This follows from the fact that since ${\mathcal{C}}$ is stable, $\Omega^{\infty}\colon\mathrm{Sp}(\mathcal{C})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{C}$ is an equivalence with inverse $\Sigma^{\infty}\colon\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Sp}(\mathcal{C})$ . ∎