free operad



A free operad is free on a collection of operations.

Given a collection kk-ary operations-to-be for each kk \in \mathbb{N}, the free operad on this collection has as nn-ary operations the collection of all trees with nn leaves equipped with a labelling of each vertex vv with a kk-ary operation, for kk the incoming edges to vv.


Let VV be a symmetric monoidal category.

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


The category of collections (Berger-Moerdijk) or 𝕊\mathbb{S}-modules (Getzler-Kapranov) of VV, or the category of VV-species, is

VColl:= nV S n. V Coll := \prod_{n \in \mathbb{N}} V^{S_n} \,.

Notice that both S 0S_0 and S 1S_1 are the trivial group.

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

U:VOperadVColl. U : V Operad \to V Coll \,.

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

F:VCollVOperad:U. F : V Coll \stackrel{\leftarrow}{\to} V Operad : U \,.

For CC a given collection, we call F(C)F(C) the operad free on the collection CC.


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


Explicit construction

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


Let 𝕋:=Core(Ω pl)\mathbb{T} := Core(\Omega_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).


  • t nΩ nt_n \in \Omega_n for the nn-corolla (the tree with a single vertex, nn inputs and its unique output root);

  • for TT any tree with nn-ary root vertex let {T i} i=1 n\{T_i\}_{i=1}^n be the sub-trees such that T=t n(T 1,,T n)T = t_n \circ (T_1, \cdots, T_n).

Then every collection KVCollK \in V Coll defines a functor K¯:𝕋 opV\bar K : \mathbb{T}^{op} \to V by the inductive formula

K¯(|)I \bar K (|) \coloneqq I
K¯(t n(T 1,,T n))K(n)K¯(T 1)K¯(T n). \bar K(t_n(T_1, \cdots, T_n)) \coloneqq K(n) \otimes \bar K(T_1) \otimes \cdots \bar K(T_n) \,.

Define moreover the functor

λ:𝕋Set \lambda : \mathbb{T} \to Set

to be the functor that sends a tree to the set of numberings of its leaves, and let λ¯:𝕋V\bar \lambda : \mathbb{T} \to V be given by postcomposition with S sSIS \mapsto \coprod_{s \in S} I, where on the right we have the coproduct of |S|{\vert S \vert} copies of the tensor unit in the monoidal category VV.

So for TT a tree with nn leaves we have

λ¯(T) σΣ nI, \bar \lambda(T) \simeq \coprod_{\sigma \in \Sigma_n} I \,,

where the coproduct ranges over the elements of the symmetric group on nn elements.


The free operad on a collection KK is isomorphic to the coend

K¯ 𝕋λ¯= T𝕋K¯(T)λ¯(T). \bar K \otimes_{\mathbb{T}} \bar \lambda = \int^{T \in \mathbb{T}} \bar K(T) \otimes \bar \lambda(T) \,.

The groupoid 𝕋\mathbb{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 [T]π 0𝕋BAut(T). \mathbb{T} \simeq \coprod_{[T] \in \pi_0\mathbb{T}} \mathbf{B} Aut(T) \,.

Therefore the above coend is equivalently

K¯ 𝕋λ¯= [T]π 0𝕋K¯(T) Aut(T)λ¯(T). \bar K \otimes_{\mathbb{T}} \bar \lambda = \coprod_{[T] \in \pi_0\mathbb{T}} \bar K(T) \otimes_{Aut(T)} \bar \lambda(T) \,.


Rooted tree operad

Let KK be the collection with K(0)=K(0) = \emptyset and K(n)=IK(n) = I for n>0n \gt 0. The corresponding free operad has as nn-ary operations all rooted trees with nn 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


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 13, 2016 12:30:07 by Mike Shulman (