Bimodule operad

Definition 19.2.1. We define a colored operad \(\oBMod \). It has three colors: \(\fa _-\), \(\fa _+\) and \(\fm \). The multimorphism sets \(\oBMod (\{X_i\}_{i \in I}; Y)\) are then given as follows:

• If \(Y = \fa _-\), then this set is empty unless \(X_i = \fa _-\) for all \(i \in I\), in which case it is the set of linear orders on \(I\);
• If \(Y = \fa _+\), then this set is empty unless \(X_i = \fa _+\) for all \(i \in I\), in which case it is the set of linear orders on \(I\);
• If \(Y = \fm \), then this set is the set of linear orders \(\{i_1 < i_2 < \dots < i_n\}\) on \(I\) with the following property: there exists precisely one index \(i_k \in I\) such that \(X_{i_k} = \fm \), \(X_{i_j} = \fa _-\) for \(j < k\) and \(X_{i_j} = \fa _+\) for \(j > k\).

Composition in \(\oBMod \) is given by composition of linear orders.