Proposition 4.14 (0NKS) · ScalingStacks

0NKS

Proposition 4.14. For any spectral category ${\mathcal{A}}$ , $\eta_{{\mathcal{A}}}\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}M{\mathcal{A}}$ is fully faithful.

0NKT

Proof. By Yoneda’s lemma, mapping spectra in $M{\mathcal{A}}$ between stably representable objects are given by mapping spectra between the representing spectrum objects, giving an equivalence

\mathrm{Map}_{M{\mathcal{A}}}(\eta_{{\mathcal{A}}}(a),\eta_{{\mathcal{A}}}(b))_{n}\simeq\map_{\Psi_{\tri}{\mathcal{A}}}(\widehat{a},{\Sigma^{n}}{\widehat{b}}).

Since $\Psi_{\tri}{\mathcal{A}}$ is a stable $\infty$ -category of spectral functors, Yoneda’s lemma also gives an equivalence

\mathrm{Map}_{{\mathcal{A}}}(a,b)_{n}\simeq\map_{\Psi_{\tri}{\mathcal{A}}}(\widehat{a},{\Sigma^{n}}{\widehat{b}}).

Hence $\mathrm{Map}_{{\mathcal{A}}}(a,b)\simeq\mathrm{Map}_{M{\mathcal{A}}}(\eta_{{\mathcal{A}}}(a),\eta_{{\mathcal{A}}}(b))$ . ∎