Omega ∞ stable operad equivalence

Proof. On underlying \(\infty \)-categories this is precisely Lemma 16.5.8. By Lemma 14.1.8, it remains to prove that \(\Omega ^\infty \) is fully faithful.

Let \(I\) be a finite set, and let \(\{F_i\colon \An _*^{\fin } \to \Oo _{\lra {1}}\}_{i \in I}\) and \(G\colon \An _*^{\fin } \to \Oo _{\lra {1}}\) be reduced excisive functors. By Theorem 18.3.5, the source of the induced map on multimorphism animae is \[ \int _{\{K_i\}_{i \in I}\in (\An _*^{\fin })^I} \Oo (\{F_i(K_i)\}_{i \in I};G(\bigwedge \nolimits _{i \in I}K_i)). \] By naturality of Theorem 18.3.5 in the evaluation operad map and the identification \(\bigwedge _{i\in I}S^0 \simeq S^0\), the map induced by \(\Omega ^\infty \) is the projection from this end to the component where all \(K_i\) are equal to \(S^0\).

We prove a slightly stronger statement by induction on the cardinality of \(I\): the same projection is an equivalence after adjoining any fixed finite tuple of spectator colors \(z_1,\dots ,z_m\) to the inputs of every multimorphism anima. For \(I=\emptyset \) there is nothing to prove. Otherwise choose \(i_0 \in I\) and put \(I' := I\setminus \{i_0\}\). By Fubini for ends, the relevant end is the end over \(K \in \An _*^{\fin }\) of \(T(K,K)\), where for \(K,L\in \An _*^{\fin }\) we define \[ T(K,L):= \int _{\{K_i\}_{i \in I'}\in (\An _*^{\fin })^{I'}} \Oo (z_1,\dots ,z_m,F_{i_0}(K),\{F_i(K_i)\}_{i \in I'};G(L\wedge \bigwedge \nolimits _{i \in I'}K_i)). \] We claim that \(T\) satisfies the hypotheses of Lemma 18.4.4. Since \(\Oo \) is stable, a reduced excisive functor \(\An _*^{\fin } \to \Oo _{\lra {1}}\) preserves finite colimits: finite colimits in \(\An _*^{\fin }\) are generated by the initial object and pushouts, and a square in a stable \(\infty \)-category is a pushout if and only if it is a pullback. Therefore \(F_{i_0}\) sends finite colimits in \(\An _*^{\fin }\) to finite operadic colimits in \(\Oo \). Since finite operadic colimits in an input variable are detected by mapping out of them, \(T(-,L)\) sends finite colimits to limits. Similarly, the functor \(L \mapsto G(L \wedge \bigwedge _{i \in I'}K_i)\) is reduced and sends pushout squares to finite limits in \(\Oo _{\lra {1}}\), and these limits are operadic. Thus \(T(K,-)\) is reduced and excisive. Ends preserve the relevant limits throughout, and the fixed spectator colors do not affect either verification.

Applying Lemma 18.4.4 collapses the \(i_0\)-variable to \(S^0\). The stronger induction hypothesis, with \(F_{i_0}(S^0)\) adjoined to the spectator colors, now collapses the remaining variables and gives an equivalence from the displayed end to \[ \Oo (\{F_i(S^0)\}_{i \in I};G(S^0)), \] which is the multimorphism anima in \(\Oo \) between the images under \(\Omega ^\infty \). This proves full faithfulness. □