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

Given $\pi_{\mathcal{O}}\colon \mathcal{O}^{\otimes} \rightarrow N(\mathrm{Fin}_*)$ a unital (∞,1)-operad, write $\mathcal{O}^{\otimes}_{\mathrm{act}} \subset \mathcal{O}^{\otimes}$ be the wide subcategory of active morphisms. Given a composable chain of active morphisms $X_0 \xrightarrow{f_1} \cdots \xrightarrow{f_n}$ with corresponding $n$-simplex $\sigma\colon \Delta^n \rightarrow \mathcal{O}^{\otimes}_{\mathrm{act}}$ and a downward-closed subset $S \subset [n]$, the (∞,1)?-category of extensions of $\sigma$ on $S$ is the full subcategroy

$$\mathrm{Ext}(\sigma,S) \subset \mathrm{Fun}(\Delta^n, \mathcal{O}^{\otimes})_{\sigma/}$$

spanned by diagrams satisfying the following properties:

a. If $i \notin S$, $g_i$ is an equivalence

a. If $i \in S$, then $g_i$ is semi-inert over an inclusion $\langle n_i \rangle \hookrightarrow \langle n_i + 1 \rangle$ which omits a single value $a_i$

a. If $1 \le i \in S$, then $\pi_{\mathcal{O}}(f'_i)$ carries $a_{i-1}$ to $a_i$

a. each $f'_i$ is active.

If the $\infty$-category of colors $\mathcal{O}$ is an ∞-groupoid, then each $\mathrm{Ext}(\sigma,S)$ is itself an ∞-groupoid, i.e. a space. A locally monoidal (∞,1)-operad can loosely be defined as one whose extension spaces satisfy excision under composition.

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; and

3. given a composable pair of morphisms $X \xrightarrow{f} Y \xrightarrow{g} Z$ with corresponding $3$-simplex $\sigma$, the associated diagram of spaces

is a (homotopy) pushout square.

This is (Lurie, def. 3.3.1.9).

The exponentiability criterion

Let $\mathrm{Ar}^{\mathrm{sInt}}(\mathcal{O}^{\otimes}) \subset \mathrm{Fun}(\Delta^1, \mathcal{O}^{\otimes})$ be the full subcategory spanned by semi-inert morphisms. The following is (Lurie, thm. 3.3.2.2).

Theorem

Suppose $\mathcal{O}^{\otimes}$ is a unital (∞,1)-operad whose $\infty$-category of colors $\mathcal{O}$ is an ∞-groupoid. Then, $\mathcal{O}^{\otimes}$ is locally monoidal if and only if the source map $s\colon \mathrm{Ar}^{\mathrm{sInt}}(\mathcal{O}^{\otimes}) \rightarrow \mathcal{O}^{\otimes}$ is an exponentiable fibration.

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