Non colored operad

Definition 12.1.1 ([May (1972); Boardman and Vogt (1973)]). A non-colored operad \(\Oo \) consists of the following data:

• A set \(\Oo (n)\) of \(n\)-ary operations, for every \(n \geq 0\);
• A composition map \[ - \circ - \colon \Oo (k) \times \prod _{i=1}^k \Oo (n_i) \to \Oo (n_1 + \dots + n_k), \quad (P,Q_1, \dots , Q_k) \mapsto P(Q_1,\dots ,Q_k) \] for all natural numbers \(k,n_1, \dots , n_k \geq 0\);
• An identity operation \(\id \in \Oo (1)\);
• A right action of the symmetric group \(\Sigma _n\) on \(\Oo (n)\) for every \(n \geq 0\).

These data are subject to the following conditions:

• Composition is unital and associative: \begin {align*} P(\id , \dots , \id ) \, = \,\,&P \, = \, \id (P), \\ R(P_1(Q^1_1, \dots , Q^1_{k_1}), \dots , P_l(Q^l_1, \dots , Q^l_{k_l})) &= (R(P_1, \dots , P_l))(Q^1_1, \dots , Q^1_{k_1},Q^2_1, \dots , Q^l_{k_l}). \end {align*}
• Composition is compatible with the permutation actions; see Chapterexercise 12.1 for details.