nLab planar operad

Idea

The notion of operad comes in two broad flavors (apart from the choice of enriching category): planar operads and symmetric operads. Accordingly, planar operads are also called non-symmetric operads. Another term is nonpermutative operads.

A planar operad is a collection (set/object in some enriching category) of $n$-ary operations for all $n \in \mathbb{N}$, equipped with a suitable notion of composition. In contrast, a symmetric operad in addition carries an action of the symmetric group $\Sigma_n$ on the object on $n$-ary operations, and all structures are required to respect this action.

The notion of planar operads takes its name from the fact that the operations in a planar operad may naturally be drawn as planar trees without, in general, a relation between two trees that cannot be related by a planar deformation into each other.

Multi-coloured planar operads over Set are equivalently known as multicategories.

Definition

In the context of (∞,1)-operads $\mathcal{O}$ exhibited by their (∞,1)-categories of operators $\mathcal{O}^\otimes$, a planar $(\infty,1)$-operad is a fibration of (∞,1)-operads

$$\mathcal{O}^\otimes \to Assoc^\otimes$$

to the associative operad/A-∞ operad. (Lurie, def. 4.1.1.6)

Properties

Relation to symmetric operads

Planar operads embed into symmetric operads by slicing over the associative operad (regarded as a symmetric operad). For more details see at Symmetric operad – Relation to planar operads.

Examples

• The operad Assoc for associative monoids is the terminal object in the category of planar $V$-operads, for choices such as $V =$ Set, sSet, Top, etc.

Related concepts

• symmetric operad

• multicategory

References

Discussion in the context of the higher algebra of (∞,1)-operads is in section 4.1.1 of

• Jacob Lurie, Higher Algebra