nLab generalized (infinity,1)-operad

Idea

An ordinary (∞,1)-operad $\mathcal{O}$ is equivalently incarnated as its (∞,1)-category of operators $\mathcal{O}^\otimes \to FinSet_*$, which in particular has the property that $\mathcal{O}^{\otimes}_{\langle 0\rangle} \simeq *$.

A generalized $(\infty,1)$-operad is like an (∞,1)-category of operators only that this condition is dropped and

$$\mathcal{C} \coloneqq \mathcal{O}^{\otimes}_{\langle 0\rangle}$$

is allowed to be any (∞,1)-category.

It is essentially the same kind of data as a family of (∞,1)-operads (see there for more) parameterized over $\mathcal{C}$.

References

Section 2.3.2 of

• Jacob Lurie, Higher Algebra