nLab locally monoidal (infinity,1)-operad

Idea

A locally monoidal $(\infty,1)$-operad (called a coherent $(\infty,1)$-operad in (Lurie)) is an (∞,1)-operad $\mathcal{O}$ whose modules over $\mathcal{O}$-algebras come equipped with a well behaved tensor product

Definition

Definition

An (∞,1)-operad $\mathcal{O}^\otimes$ is locally monoidal if

1. it is unital;

2. the underlying (∞,1)-category $\mathcal{O}$ is an ∞-groupoid

3. (some third condition).

This is (Lurie, def. 3.3.1.9).

Examples

Locally monoidal $(\infty,1)$-operads include

• Comm

  (Lurie, example 3.3.1.12)

• Assoc

  (Lurie, prop. 4.1.1.16)

• little 0-cubes operad

  (Lurie, example 3.3.1.13)

• little k-cubes operad for all $k \in \mathbb{N}$

  (Lurie, theorem 5.1.1.1)

References

Section 3.3.1 of

• Jacob Lurie, Higher Algebra