Algebras and modules
Model category presentations
Geometry on formal duals of algebras
A free operad is free on a collection of operations.
Given a collection -ary operations-to-be for each , the free operad on this collection has as -ary operations the collection of all trees with leaves equipped with a labelling of each vertex with a -ary operation, for the incoming edges to .
Let be a symmetric monoidal category.
For a discrete group, write for the category of objects of equipped with a -action. For symmetric monoidal this is again a symmetric monoidal category and the forgetful functor is symmetric monoidal.
The category of collections (Berger-Moerdijk) or -modules (Getzler-Kapranov) of , or the category of -species, is
Notice that both and are the trivial group.
So a -operad is a special -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 a given collection, we call the operad free on the collection .
The free operad functor may more explcitly be described as follows (see (Berger-Moerdijk, section 5.8)).
Let be the core of the category of planar rooted trees and non-planar morphisms (so the morphisms need not respect the given planar structure).
for the -corolla (the tree with a single vertex, inputs and its unique output root);
for any tree with -ary root vertex let be the sub-trees such that .
Then every collection defines a functor 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 be given by postcomposition with , where on the right we have the coproduct of copies of the tensor unit in the monoidal category .
So for a tree with leaves we have
where the coproduct ranges over the elements of the symmetric group on elements.
The free operad on a collection is isomorphic to the coend
Rooted tree operad
Let be the collection with and for . The corresponding free operad has as -ary operations all rooted trees with 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