Spectrum objects form a stable ∞-category

Proof. Given \(X\in D\), Lemma 4.3.12 gives a pointed left exact functor \[ X^{(-)}\colon (\An _*^{\fin })\catop \to D \] satisfying \(X^{S^0}\simeq X\). Since \(X^{(-)}\) sends pushouts in \(\An _*^{\fin }\) to pullbacks in \(D\), we have \[ X^{S^{n+1}} \simeq X^{\Sigma S^n} \simeq \Omega X^{S^n} \] for every \(n\geq 0\). Thus the comparison automorphism of \(\Omega ^3X\) is induced by the analogous automorphism of \(S^3\) in \(\An _*^{\fin }\).

It remains to prove the claim in \(\An _*^{\fin }\) for the generator \(X = S^0\). We must show that the composite \[ \tau \colon S^3 \simeq \Sigma (\Sigma ^2S^0) \xrightarrow {\cong } \Sigma ^2(\Sigma S^0) \simeq S^3 \] is pointed homotopic to the identity. By Remark 2.4.20, it suffices to show that \(\tau \) has degree \(1\). Using the swap map \(\sigma _{X,Y}\) from Definition 2.4.10, we may identify \(\tau \) with \(\sigma _{S^1,S^2}\). The isomorphism \(S^2 \cong S^1 \wedge S^1\) from Lemma 2.4.16 then gives a decomposition \[ S^3 \simeq S^1\wedge S^1\wedge S^1 \xrightarrow {\id \wedge \sigma _{S^1,S^1}} S^1\wedge S^1\wedge S^1 \xrightarrow {\sigma _{S^1,S^1}\wedge \id } S^1\wedge S^1\wedge S^1 \simeq S^3. \] Both maps in this decomposition are suspensions of \(\sigma _{S^1,S^1}\), up to the natural identifications of smash products with suspensions. Their degrees are therefore equal to that of \(\sigma _{S^1,S^1}\) by Remark 2.4.20. Since this swap map is an isomorphism, its degree is \(\pm 1\). We conclude that \[ \deg (\tau )=\deg (\id \wedge \sigma _{S^1,S^1})\cdot \deg (\sigma _{S^1,S^1}\wedge \id )=\deg (\sigma _{S^1,S^1})^2=1, \] as desired. The construction is natural in \(X\), since the comparison is induced by the automorphism \(\tau \) of \(S^3\) under the equivalence of Lemma 4.3.12. □