Operad gives ∞ operad

Construction 12.4.1. Let \(\Oo \) be a colored operad. We describe a (2,1)-category \(\Oo ^{\otimes }\) as follows:

• The objects of \(\Oo ^{\otimes }\) are unordered tuples \(\{x_i\}_{i \in I}\);
• A morphism \(\{x_i\}_{i \in I} \to \{y_j\}_{j \in J}\) in \(\Oo ^{\otimes }\) consists of a span \[ I \xleftarrow {f} K \xrightarrow {g} J \] of finite sets together with a multimorphism \(\phi _j \in \Oo (\{x_{f(k)}\}_{k \in K_j};y_j)\) for every \(j \in J\); here we write \(K_j := g^{-1}(j)\) for the preimage of \(j\) under \(g\). Together with the canonical notion of isomorphisms between such labeled spans, these form a groupoid \(\Hom _{\Oo ^{\otimes }}(\{x_i\}_{i \in I}, \{y_j\}_{j \in J})\).
• The identity of \(\{x_i\}_{i \in I}\) is given by taking \(f = g = \id _I \colon I \to I\), and taking \(\phi _j = \id _{x_j} \in \Oo (\{x_i\}_{i \in \{j\}};x_j)\).
• The composition of \((f,g,\phi _j)\colon \{x_i\}_{i \in I} \to \{y_j\}_{j \in J}\) and \((h,k, \psi _l)\colon \{y_j\}_{j \in J} \to \{z_l\}_{l \in L}\) is given by \((f\circ \pr _K,k \circ \pr _M, \xi _l)\) where \(\pr _K\) and \(\pr _M\) are the projection maps in the following pullback diagram

  

  and for every \(l \in L\) the multimorphism \(\xi _l \in \Oo (\{x_{f(u)}\}_{(u,m) \in (K \times _J M)_l};z_l)\) is the image of \((\psi _l, (\phi _{h(m)})_{m \in M_l})\) under the composition map \[ \circ \colon \Oo (\{y_{h(m)}\}_{m \in M_l};z_l) \times \prod _{m \in M_l} \Oo (\{x_{f(u)}\}_{u \in K_{h(m)}}; y_{h(m)}) \to \Oo (\{x_{f(u)}\}_{(u,m) \in (K \times _J M)_l};z_l). \]
• The operad axioms make composition unital and associative up to the canonical isomorphisms induced by pullbacks of spans.