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locally monoidal (infinity,1)-operad

Idea

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

Definition

Definition

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

1. it is unital;

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

3. (some third condition).

This is (Lurie, def. 3.3.1.9).

Examples

Locally monoidal $(\mathrm{\infty },1)$-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