### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

A free operad is free on a collection of operations.

Given a collection $k$-ary operations-to-be for each $k\in ℕ$, the free operad on this collection has as $n$-ary operations the collection of all trees with $n$ leaves equipped with a labelling of each vertex $v$ with a $k$-ary operation, for $k$ the incoming edges to $v$.

## Definition

Let $V$ be a symmetric monoidal category.

For $G$ a discrete group, write ${V}^{G}$ for the category of objects of $V$ equipped with a $G$-action. For $V$ symmetric monoidal this is again a symmetric monoidal category and the forgetful functor ${V}^{G}\to V$ is symmetric monoidal.

###### Definition

The category of collections (Berger-Moerdijk) or $𝕊$-modules (Getzler-Kapranov) of $V$, or the category of $V$-species, is

$V\mathrm{Coll}:=\prod _{n\in ℕ}{V}^{{S}_{n}}\phantom{\rule{thinmathspace}{0ex}}.$V Coll := \prod_{n \in \mathbb{N}} V^{S_n} \,.

Notice that both ${S}_{0}$ and ${S}_{1}$ are the trivial group.

So a $V$-operad $P$ is a special $V$-collection with extra structure relating its components. This gives an evident forgetful functor

$U:V\mathrm{Operad}\to V\mathrm{Coll}\phantom{\rule{thinmathspace}{0ex}}.$U : V Operad \to V Coll \,.
###### Definition

The free functor left adjoint to this forgetful functor is the the free operad functor

$F:V\mathrm{Coll}\stackrel{←}{\to }V\mathrm{Operad}:U\phantom{\rule{thinmathspace}{0ex}}.$F : V Coll \stackrel{\leftarrow}{\to} V Operad : U \,.

For $C$ a given collection, we call $F\left(C\right)$ the operad free on the collection $C$.

###### Remark

This free/forgetful adjunction is used to define the model structure on operads by transfer.

## Properties

### Explicit construction

The free operad functor may more explcitly be described as follows (see (Berger-Moerdijk, section 5.8)).

###### Definition

Let $𝕋:=\mathrm{Core}\left({\Omega }_{\mathrm{pl}}\right)$ be the core of the category of planar rooted trees and non-planar morphisms (so the morphisms need not respect the given planar structure).

Write

• ${t}_{n}\in {\Omega }_{n}$ for the $n$-corolla (the tree with a single vertex, $n$ inputs and its unique output root);

• for $T$ any tree with $n$-ary root vertex let $\left\{{T}_{i}{\right\}}_{i=1}^{n}$ be the sub-trees such that $T={t}_{n}\circ \left({T}_{1},\cdots ,{T}_{n}\right)$.

Then every collection $K\in V\mathrm{Coll}$ defines a functor $\overline{K}:{𝕋}^{\mathrm{op}}\to V$ by the inductive formula

$\overline{K}:\mid ↦I$\bar K : | \mapsto I
$\overline{K}:T↦\overline{K}\left({t}_{n}\left({T}_{1},\cdots ,{T}_{n}\right)\right):=K\left(n\right)\otimes K\left({T}_{1}\right)\otimes \cdots K\left({T}_{n}\right)\phantom{\rule{thinmathspace}{0ex}}.$\bar K : T \mapsto \bar K(t_n(T_1, \cdots, T_n)) := K(n) \otimes K(T_1) \otimes \cdots K(T_n) \,.

Define moreover the functor

$\lambda :𝕋\to \mathrm{Set}$\lambda : \mathbb{T} \to Set

to be the functor that sends a tree to the set of numberings of its leaves, and let $\overline{\lambda }:𝕋\to V$ be given by postcomposition with $S↦{\coprod }_{s\in S}I$, where on the right we have the coproduct of $\mid S\mid$ copies of the tensor unit in the monoidal category $V$.

So for $T$ a tree with $n$ leaves we have

$\overline{\lambda }\left(T\right)\simeq \coprod _{\sigma \in {\Sigma }_{n}}I\phantom{\rule{thinmathspace}{0ex}},$\bar \lambda(T) \simeq \coprod_{\sigma \in \Sigma_n} I \,,

where the coproduct ranges over the elements of the symmetric group on $n$ elements.

###### Proposition

The free operad on a collection $K$ is isomorphic to the coend

$\overline{K}{\otimes }_{𝕋}\overline{\lambda }={\int }^{T\in 𝕋}\overline{K}\left(T\right)\otimes \overline{\lambda }\left(T\right)\phantom{\rule{thinmathspace}{0ex}}.$\bar K \otimes_{\mathbb{T}} \bar \lambda = \int^{T \in \mathbb{T}} \bar K(T) \otimes \bar \lambda(T) \,.
###### Remark

The groupoid $𝕋$ is equivalent to the disjoint union over isomorphism classes of planar trees of the one-object groupoids with morphisms the given automorphism group

$𝕋\simeq \coprod _{\left[T\right]\in {\pi }_{0}𝕋}B\mathrm{Aut}\left(T\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{T} \simeq \coprod_{[T] \in \pi_0\mathbb{T}} \mathbf{B} Aut(T) \,.

Therefore the above coend is equivalently

$\overline{K}{\otimes }_{𝕋}\overline{\lambda }=\coprod _{\left[T\right]\in {\pi }_{0}𝕋}\overline{K}\left(T\right){\otimes }_{\mathrm{Aut}\left(T\right)}\overline{\lambda }\left(T\right)\phantom{\rule{thinmathspace}{0ex}}.$\bar K \otimes_{\mathbb{T}} \bar \lambda = \coprod_{[T] \in \pi_0\mathbb{T}} \bar K(T) \otimes_{Aut(T)} \bar \lambda(T) \,.

## Examples

Let $K$ be the collection with $K\left(0\right)=\varnothing$ and $K\left(n\right)=I$ for $n>0$. The corresponding free operad has as $n$-ary operations all rooted trees with $n$ leaves. And composition of operations is given by grafting of trees.

Riemann surfaces operad (TO BE EXPANDED)

Deligne-Mumford opeard (TO BE EXPANDED)

little discs operad, framed little discs operad (TO BE EXPANDED) – See Deligne conjecture

## References

A brief remark on free operads is in (1.12) of

• Ezra Getzler, Mikhail Kapranov, Cyclic operads and cyclic homology, Conf. Proc. Lect. Notes Geom. Topology IV, Int. Press Camb. (1995), 167–201.

A detailed discussion is in Part II, chapter I, section 1.9 of

• Martin Markl, S. Shnider, Jim Stasheff, Operads in Algebra, Topology and Physics, Surveys and Monographs of the Amer. Math. Soc. 96 (2002).

and in section 3 of

The coend-description is given in section 5.8 of

Revised on March 6, 2012 19:29:44 by Todd Trimble (67.80.8.47)