nLab family of (infinity,1)-operads

Definition

For the following we model (∞,1)-operads $\mathcal{O}$ specifically by their (∞,1)-categories of operators $\mathcal{O}^\otimes$ and these specifically as quasi-categories.

Let $\mathcal{C}$ be an (∞,1)-category. A $\mathcal{C}$-family of (∞,1)-operads, is a fibration in the model structure for quasi-categories

$$p \colon \mathcal{O}^{\otimes} \to \mathcal{C} \times FinSet_*$$

such that

• For $C \in \mathcal{C}$ any object, for $X \in \mathcal{O}^\otimes_C$ over $\langle n\rangle \in FinSet_*$, and for $\alpha \colon \langle n\rangle \to \langle k\rangle$ any inert morphism, then there exists a $p$-coCartesian morphism $\overline \alpha \colon X \to Y$ in $\mathcal{O}^\otimes_C$.

(Lurie, 2.3.2.10)

Properties

Relation to generalized $(\infty,1)$-operads

For $\mathcal{C}$ an (∞,1)-category, a fibration $\mathcal{O}^\otimes \to \mathcal{C} \times FinSet_*$ in the model structure for quasi-categories is a $\mathcal{C}$-family of $(\infty,1)$-operads precisely if it is a fibration of generalized (∞,1)-operads such that the underlying map $\mathcal{O}^\otimes_{\langle 0\rangle} \to \mathcal{C}$ is an acyclic Kan fibration.

References

• Jacob Lurie, Higher Algebra