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free operad

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Contents

Idea

A free operad is free on a collection of operations.

Given a collection k-ary operations-to-be for each k∈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→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

VColl:=∏ n∈ℕV 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:VOperad→VColl.U : V Operad \to V Coll \,.
Definition

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

F:VColl→←VOperad: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:=Core(Ω 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∈Ω 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∘(T 1,⋯,T n).

Then every collection K∈VColl defines a functor K¯:𝕋 op→V by the inductive formula

K¯:∣↦I\bar K : | \mapsto I
K¯:T↦K¯(t n(T 1,⋯,T n)):=K(n)⊗K(T 1)⊗⋯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

λ:𝕋→Set\lambda : \mathbb{T} \to Set

to be the functor that sends a tree to the set of numberings of its leaves, and let λ¯:𝕋→V be given by postcomposition with S↦∐ s∈SI, where on the right we have the coproduct of ∣S∣ copies of the tensor unit in the monoidal category V.

So for T a tree with n leaves we have

λ¯(T)≃∐ σ∈Σ nI,\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

K¯⊗ 𝕋λ¯=∫ T∈𝕋K¯(T)⊗λ¯(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≃∐ [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) \,.

Examples

Rooted tree operad

Let K be the collection with K(0)=∅ 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.

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)
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