Colored operad

Definition 12.2.1. A colored operad (or multicategory) \(\Oo \) consists of the following structure:

• There is a set \(\Oo ^{\simeq }\) of colors (sometimes also called the objects of \(\Oo \));
• For colors \(x, y_1, \dots , y_n \in \Oo ^{\simeq }\), there is a set \(\Oo ((y_1, \dots , y_n); x)\) of multimorphisms from \((y_1, \dots , y_n)\) to \(x\). We refer to \(x\) as the output color and to the \(y_i\) as the input colors;
• If we are additionally given colors \(z^i_j \in \Oo ^{\simeq }\) for \(i = 1, \dots , n\) and \(j = 1, \dots , k_i\), there is a composition map \[ \circ \colon \Oo ((y_1, \dots , y_n); x) \times \prod _{i=1}^n \Oo ((z^i_1, \dots , z^i_{k_i}); y_i) \to \Oo ((z^1_1, \dots , z^1_{k_1}, \dots , z^n_1, \dots , z^n_{k_n});x). \]
• For every color \(x\) there is an identity operation \(\id _x \in \Oo ((x);x)\);
• Composition should be unital and associative like before;
• For every permutation \(\sigma \in \Sigma _n\) there is a ‘permutation isomorphism’ \[ \Oo ((y_1, \dots , y_n); x) \iso \Oo ((y_{\sigma (1)}, \dots , y_{\sigma (n)}); x), \qquad P \mapsto P\sigma , \] which is compatible with the multiplication in the group \(\Sigma _n\) and is furthermore compatible with composition in \(\Oo \). We leave a precise formulation of this condition to the reader; see Chapterexercise 12.1.