Module operad

Example 12.2.8 (Module operad). There is the module operad \(\oMod \) which has two colors \(a\) and \(m\), so that \(\oMod ^{\simeq } = \{a, m\}\). The operations are given as follows:

• If the output color is \(a\), the operations should encode a commutative algebra structure, so we only want to allow operations all of whose input colors are \(a\) as well: \[ \oMod ((x_1, \dots , x_n); a) := \begin {cases} * & \text {if $x_j = a$ for all $j=1,\dots ,n$,} \\ \emptyset & \text {otherwise}, \end {cases} \]
• If the output color is \(m\), the operations should encode the action of the commutative algebra on a module, so we only want operations that have precisely one input color that is \(m\), while all others are \(a\): \[ \oMod ((x_1, \dots , x_n); m) := \begin {cases} * & \text {if for some $i$ we have $x_i = m$ and $x_j = a$ for $j \neq i$}, \\ \emptyset & \text {otherwise}. \end {cases} \]

The composition is uniquely determined.

Note that restricting the operations to the color \(a\) gives the commutative operad, while restricting the operations to the color \(m\) gives the trivial operad: \[ \oMod _a \cong \Comm \qquadtext {and} \oMod _m \cong \Triv . \]