nLab d-operad

Idea

$d$-operads are to ∞-operads as $d$-categories are to ∞-categories; just as $d$-categories are simply $\infty$-categories whose mapping spaces are $(d-1)$-truncated, $d$-operads are simply $\infty$-categories whose (multi)morphism spaces are $(d-1)$-truncated.

Definitions

Definition

Fix $d \in \mathbb{N}$ a natural number An $\infty$-operad $\mathcal{O}^{\otimes}$ is essentially $d$ (or a $d$-operad) if, for all colors $X_1,\dots,X_n,Y \in \mathcal{O}$, the space $\mathrm{Mul}_{\mathcal{O}}(X_1,\dots,X_n;Y)$ is $(d-1)$-truncated

Let $\mathrm{Op}_d \subset \mathrm{Op}_{\infty}$ be the full subcategory spanned by d-operads.

Proposition

$\mathrm{Op}_d \subset \mathrm{Op}_\infty$ is a localizing subcategory.

We refer to the localization functor $h_d:\mathrm{Op}_\infty \rightarrow \mathrm{Op}_d$ as the homotopy $d$-operad functor.

Properties

Proposition

A symmetric monoidal ∞-category is essentially $d$ when regarded as an ∞-operad if and only if its underlying $\infty$-category is a $d$-category.

Proposition

An $\infty$-operad $\mathcal{O}^{\otimes}$ is $d+1$-connected if and only if the canonical map $h_{d} \mathcal{O}^{\otimes} \rightarrow \mathbb{E}_\infty^{\otimes}$ is an equivalence.

Examples

• $\mathbb{E}_0$ and $\mathbb{E}_\infty$ are 0-operads.
• the operadic nerve of an operad is a $1$-operad.
• $\mathbb{E}_2$ is a $2$-operad.

Related concepts

• ∞-operad

References

• Tomer Schlank, Lior Yanovski: On d-Categories and d-Operads (arXiv:1902.04061v1)