Contents

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

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

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

## Properties

• 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.