nLab
free operad

Idea

A free operad is free on a collection of operations.

Given a collection $k$-ary operations-to-be for each $k\in \mathbb{N}$, 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

$$V\mathrm{Coll}:=\prod _{n\in \mathbb{N}}{V}^{S_n}.$$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

$$U:V\mathrm{Operad}\to V\mathrm{Coll}.$$U : V Operad \to V Coll \,.

Definition

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

$$F:V\mathrm{Coll}\overleftarrow{\to }V\mathrm{Operad}:U.$$F : V Coll \stackrel{\leftarrow}{\to} V Operad : U \,.

For $C$ a given collection, we call $F(C)$ 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 $𝕋:=\mathrm{Core}({\Omega }_{\mathrm{pl}})$ 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 $\{T_i{\}}_{i=1}^n$ be the sub-trees such that $T=t_n\circ (T_1,\cdots ,T_n)$.

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

$$\overline{K}:\mid \mapsto I$$\bar K : | \mapsto I

$$\overline{K}:T\mapsto \overline{K}(t_n(T_1,\cdots ,T_n)):=K(n)\otimes K(T_1)\otimes \cdots K(T_n).$$\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 $S\mapsto {\coprod }_{s\in S}I$, 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

$$\overline{\lambda }(T)\simeq \coprod _{\sigma \in {\Sigma }_n}I,$$\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

$$\overline{K}{\otimes }_{𝕋}\overline{\lambda }={\int }^{T\in 𝕋}\overline{K}(T)\otimes \overline{\lambda }(T).$$\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

$$𝕋\simeq \coprod _{[T]\in {\pi }_0𝕋}B\mathrm{Aut}(T).$$\mathbb{T} \simeq \coprod_{[T] \in \pi_0\mathbb{T}} \mathbf{B} Aut(T) \,.

Therefore the above coend is equivalently

$$\overline{K}{\otimes }_{𝕋}\overline{\lambda }=\coprod _{[T]\in {\pi }_0𝕋}\overline{K}(T){\otimes }_{\mathrm{Aut}(T)}\overline{\lambda }(T).$$\bar K \otimes_{\mathbb{T}} \bar \lambda = \coprod_{[T] \in \pi_0\mathbb{T}} \bar K(T) \otimes_{Aut(T)} \bar \lambda(T) \,.

Examples

Rooted tree operad

Let $K$ be the collection with $K(0)=\varnothing$ and $K(n)=I$ 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.

Related concepts

• free category, free monoid

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

• Clemens Berger, Ieke Moerdijk, The Boardman-Vogt resolution of operads in monoidal model categories , Topology 45 (2006), 807–849. (pdf)

The coend-description is given in section 5.8 of

• Clemens Berger, Ieke Moerdijk, Axiomatic homotopy theory for operads (arXiv:0206094)