Left module operad

Example 12.2.9 (Left module operad). There is also a non-commutative version of the module operad, capturing the structure of a left module over an associative algebra. We will denote it by \(\oLMod \) and call it the left module operad. It again has two colors, \(\oLMod ^{\simeq } = \{a,m\}\). The operations are defined as follows:

• If the output color is \(a\), we again only want operations all of whose input colors are \(a\), but we should allow them to be ordered arbitrarily: \[ \oLMod ((x_1, \dots , x_n); a) := \begin {cases} \{\text {linear orders on }\{1,\dots ,n\}\} & \text {if } x_j = a \text { for all } j=1, \dots , n, \\ \emptyset & \text {otherwise}. \end {cases} \]
• If the output color is \(m\), we should again only allow operations such that precisely one of the inputs is \(m\). To allow for permutations of the other inputs, let us write \(I_m := \{ i \in \{1,\dots ,n\} \mid x_i = m \}\) for the set of indices corresponding to module inputs, and let \(I_a := \{ j \in \{1,\dots ,n\} \mid x_j = a \}\) be the set of indices corresponding to algebra inputs. We then set \[ \oLMod ((x_1, \dots , x_n); m) := \begin {cases} \{\text {linear orders on }I_a\} & \text {if } |I_m| = 1 \text { (which implies } |I_a| = n-1 \text {)}, \\ \emptyset & \text {if } |I_m| \neq 1. \end {cases} \] Note that this set has cardinality \((n-1)!\).

Composition substitutes an ordered list into each algebra input and concatenates the resulting lists in the order specified by the outer operation. For an operation with output color \(m\), the ordered lists substituted into the algebra inputs are followed by the ordered list of algebra inputs belonging to the operation substituted into the unique module input; the unique module input itself is carried along. The symmetric group action transports these linear orders along the induced bijections between the sets of algebra input positions. These rules are unital, associative and compatible with the symmetric group actions.

As in the commutative case, restricting operations to the color \(a\) recovers the underlying algebra operad, while restricting to the color \(m\) yields the trivial operad: \[ \oLMod _a \cong \Assoc \qquad \text { and } \qquad \oLMod _m \cong \Triv . \]