Lemma 14.2.8. Let \(\Oo \) be an \(\infty \)-operad. Assume that the following two conditions are satisfied:
- (1)
-
For every object \(\{x_i\}_{i \in I}\) of \(\Oo ^{\otimes }\) the functor \[ \Oo (\{x_i\}_{i \in I}; -)\colon \Oo _{\lra {1}} \to \An \] is corepresentable, in the sense that there exists a multimorphism \(\phi \colon \{x_i\}_{i \in I} \to y\) such that for every other color \(z \in \Oo ^{\simeq }\) precomposition with \(\phi \) induces an equivalence \[ \Hom _{\Oo _{\lra {1}}}(y,z) \iso \Oo (\{x_i\}_{i \in I}; z). \] This condition uniquely determines \(y\), and we will denote it by \(\bigotimes _{i \in I} x_i\).
- (2)
-
For every morphism \(f\colon I \to J\) of finite sets, the canonical comparison map \[ \bigotimes _{i \in I} x_i \to \bigotimes _{j \in J} \bigotimes _{i \in f^{-1}(j)} x_i \] is an equivalence.
Then \(p_{\Oo }\colon \Oo ^{\otimes } \to \Span (\Fin )\) is a cocartesian fibration, corresponding to a symmetric monoidal structure on the underlying \(\infty \)-category \(\Oo ^{\otimes }_{\lra {1}}\).
Proof. We need to produce cocartesian lifts of morphisms in \(\Span (\Fin )\). Cocartesian lifts of backwards morphisms exist by the defining property of \(\infty \)-operads, so it suffices to produce cocartesian lifts for active spans \(\alpha = (I \xleftarrow {=} I \xrightarrow {f} J)\). Let \(X = \{x_i\}_{i \in I}\) be an object in \(\Oo ^{\otimes }_I\). Applying the assumption on \(\Oo \) to each \(X_j := \{x_i\}_{i \in f^{-1}(j)}\), we may find multimorphisms \(\phi _j\colon \{x_i\}_{i \in f^{-1}(j)} \to y_j\) in \(\Oo \) corepresenting the functor \(\Oo (\{x_i\}_{i \in f^{-1}(j)}; -)\colon \Oo _{\lra {1}} \to \An \). The multimorphisms \(\phi _j\) define a morphism \(\phi \colon X \to Y := \{y_j\}_{j \in J}\) in \(\Oo ^{\otimes }\). We wish to show that \(\phi \) is a cocartesian lift of \(\alpha \).
To this end, consider any other span \(\beta = (J \xleftarrow {g} K \xrightarrow {h} L)\) of finite sets, and consider an object \(Z = \{z_l\}_{l \in L}\) of \(\Oo ^{\otimes }_L\). We need to show that precomposition with \(\phi \) induces an equivalence \[ - \circ \phi \colon \Hom _{\Oo ^{\otimes }}^{\beta }(Y,Z) \iso \Hom _{\Oo ^{\otimes }}^{\beta \circ \alpha }(X,Z). \] Since \(Z\) is a product in \(\Oo ^{\otimes }\) of the one-element tuples \((z_l)\), we may assume that \(L = \lra {1}\), so that there is only a single color \(z := z_1\). Using the inert morphisms \(\{y_j\} \to \{y_{g(k)}\}_{k \in K}\) and \(\{x_i\}_{(i,k) \in I \times _J K}\), we must equivalently show that composition with the map \(\phi \) induces an equivalence \[ - \circ \phi \colon \Oo (\{y_{g(k)}\}_{k \in K}; z) \iso \Oo (\{x_i\}_{(i,k) \in I \times _J K}; z). \] Now, applying condition (1) another time to the families \(\{y_{g(k)}\}_{k \in K}\) and \(\{x_i\}_{(i,k) \in I \times _J K}\), both sides are corepresented by objects \(\bigotimes _{k \in K} y_{g(k)}\) and \(\bigotimes _{(i,k) \in I \times _J K} x_i\), respectively. But since \(y_{g(k)} = \bigotimes _{i \in f^{-1}(g(k))} x_i\), the claim now follows from the equivalence \[ \bigotimes _{(i,k) \in I \times _J K} x_i \iso \bigotimes _{k \in K} \bigotimes _{i \in f^{-1}(g(k))} x_i \] obtained by applying condition (2) to the projection map \(I \times _J K \to K\). Under the two corepresenting equivalences, the map induced by composition with \(\phi \) is precisely precomposition with this comparison map, by the definition of the latter through operadic composition with the multimorphisms \(\phi _j\). Thus \(\phi \) is cocartesian.
We have now produced cocartesian lifts of all active spans. Together with the inert lifts and the factorization of Remark 14.1.12, these make \(p_{\Oo }\) a cocartesian fibration. Since \(\Oo \) is already an \(\infty \)-operad, Lemma 14.2.4 shows that its straightening preserves finite products, and hence defines the asserted symmetric monoidal structure. □
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