symmetric monoidal (∞,1)-category of spectra
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 $v$.
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.
The category of collections (Berger-Moerdijk) or $\mathbb{S}$-modules (Getzler-Kapranov) of $V$, or the category of $V$-species, is
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
The free functor left adjoint to this forgetful functor is the the free operad functor
For $C$ a given collection, we call $F(C)$ the operad free on the collection $C$.
This free/forgetful adjunction is used to define the model structure on operads by transfer.
The free operad functor may more explcitly be described as follows (see (Berger-Moerdijk, section 5.8)).
Let $\mathbb{T} := Core(\Omega_pl)$ be the core of the category of planar rooted trees and non-planar morphisms (so the morphisms need not respect the given planar structure).
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 $K \in V Coll$ defines a functor $\bar K : \mathbb{T}^{op} \to V$ by the inductive formula
Define moreover the functor
to be the functor that sends a tree to the set of numberings of its leaves, and let $\bar \lambda : \mathbb{T} \to V$ be given by postcomposition with $S \mapsto \coprod_{s \in S} I$, where on the right we have the coproduct of ${\vert S \vert}$ copies of the tensor unit in the monoidal category $V$.
So for $T$ a tree with $n$ leaves we have
where the coproduct ranges over the elements of the symmetric group on $n$ elements.
The free operad on a collection $K$ is isomorphic to the coend
The groupoid $\mathbb{T}$ is equivalent to the disjoint union over isomorphism classes of planar trees of the one-object groupoids with morphisms the given automorphism group
Therefore the above coend is equivalently
Let $K$ be the collection with $K(0) = \emptyset$ and $K(n) = I$ for $n \gt 0$. 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
A brief remark on free operads is in (1.12) of
A detailed discussion is in Part II, chapter I, section 1.9 of
and in section 3 of
The coend-description is given in section 5.8 of