nLab d-operad

Content

Content

Idea

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

Definitions

Definition

Fix dd \in \mathbb{N} a natural number An \infty -operad 𝒪 \mathcal{O}^{\otimes} is essentially dd (or a dd-operad) if, for all colors X 1,,X n,Y𝒪X_1,\dots,X_n,Y \in \mathcal{O}, the space Mul 𝒪(X 1,,X n;Y)\mathrm{Mul}_{\mathcal{O}}(X_1,\dots,X_n;Y) is ( d 1 ) (d-1) -truncated

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

Proposition

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

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

Properties

Proposition

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

Proposition

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

Examples

References

Last revised on July 15, 2024 at 13:51:46. See the history of this page for a list of all contributions to it.