planar operad



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 nn-ary operations for all nn \in \mathbb{N}, equipped with a suitable notion of composition. In contrast, a symmetric operad in addition carries an action of the symmetric group Σ n\Sigma_n on the object on nn-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.


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

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

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


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.



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

Revised on April 7, 2015 16:24:56 by Noam Zeilberger (