From colored to ∞ operads

Proof. Define a classical 1-category \(\Oo ^{\otimes ,\mathrm {Lurie}}\) as in Construction 12.4.1, with the additional requirement that in the underlying span \(I \xleftarrow {f} K \xrightarrow {g} J\) the finite set \(K\) is an actual subset of \(I\) and \(f\) is its inclusion. Composition is defined as in Construction 12.4.1, taking as the pullback of spans the literal preimage: the composite of \((K \subseteq I, g\colon K \to J)\) with \((M \subseteq J, k\colon M \to L)\) has underlying subset \(g^{-1}(M) \subseteq K \subseteq I\) and underlying map the restriction of \(k \circ g\) to it. This is again a subset inclusion, and the resulting composition is strictly associative: composing with a third morphism \((N \subseteq L, l\colon N \to P)\) in either of the two possible orders yields the subset \(g^{-1}(k^{-1}(N)) \subseteq I\), together with the restriction of \(l \circ k \circ g\). The identity of \(\{x_i\}_{i \in I}\) is given by \(K = I\) and \(g = \id _I\), labeled by identity multimorphisms. On the level of the multimorphism labels, associativity and unitality of this composition are precisely the associativity and unitality axioms of the colored operad \(\Oo \). Thus \(\Oo ^{\otimes ,\mathrm {Lurie}}\) is a classical 1-category. Sending \(\{x_i\}_{i \in I}\) to \(I_+\) and a morphism \((K \subseteq I, g, \{\phi _j\}_{j \in J})\) to the pointed map \(I_+ \to J_+\) that restricts to \(g\) on \(K\) and sends \(I \setminus K\) to the basepoint defines a functor of 1-categories \[ p_{\Oo }\colon \Oo ^{\otimes ,\mathrm {Lurie}} \longrightarrow \Fin _*. \] Its fiber over \(I_+\) consists of the \(I\)-tuples of colors of \(\Oo \) and the \(I\)-tuples of unary multimorphisms between them, i.e. it is the \(I\)-fold power of the category of colors and unary multimorphisms of \(\Oo \). For an inert morphism, the morphism obtained by restricting an \(I\)-tuple of colors and labeling the resulting unary operations by identities is \(p_{\Oo }\)-cocartesian. These cocartesian lifts give the required equivalences from the fiber over \(I_+\) to the product of the fibers over its elements.

It remains to verify the condition on morphism sets. Let \(X=\{x_i\}_{i\in I}\) and \(Y=\{y_j\}_{j\in J}\). For a pointed map \(\alpha \colon I_+\to J_+\), the set of morphisms \(X\to Y\) over \(\alpha \) is, by construction, \[ \prod _{j\in J} \Oo \bigl ((x_i)_{i\in \alpha ^{-1}(j)};y_j\bigr ). \] The \(j\)-th factor is precisely the set of morphisms from \(X\) to \(y_j\) over the composite of \(\alpha \) with the Segal map \(\rho _j\). Consequently the square in condition (2) of Lurie’s definition is a pullback. Thus \(p_{\Oo }\) is an \(\infty \)-operad in Lurie’s sense. A morphism of colored operads acts on tuples and multimorphism labels and thereby induces a functor between the resulting categories over \(\Fin _*\). Applying an inverse to the equivalence in Proposition 17.4.8 therefore produces the claimed functor from colored operads to \(\infty \)-operads over \(\Span (\Fin )\). □