Trivial operad on a 1-category

Example 12.2.6 (Trivial operad on a 1-category). Let \(C\) be a 1-category. We define a colored operad \(\Triv _C\), called the trivial operad generated by \(C\), as follows: its colors are the objects of \(C\), and its multimorphisms are given by \[ \Triv _C((y_1, \dots , y_n);x) \quad := \quad \begin {cases} \Hom _C(y_1,x) & \text {if $n = 1$,} \\ \emptyset & \text {otherwise.} \end {cases} \] The identity operations are the identity morphisms of \(C\), the composition is the composition in \(C\), and the permutation operations are trivial since \(\Sigma _1\) is the trivial group. In other words, \(\Triv _C\) has no operations besides the unary ones, and these are precisely the morphisms of \(C\).

The trivial operad \(\Triv \) is the special case \(C = *\), while \(C = \emptyset \) gives the initial colored operad from the warning above.