5 Operad and cooperad structures
We now describe an operad structure on and the corresponding dual cooperad structure on .
Given an -tree, we can produce a bracket expression. For example, we send
We can then send forests to products of bracket expressions, e.g.
The Jacobi identity translates to the usual Jacobi identity of the Lie/Poisson bracket. With the bracket expressions, we can interpret as operad in the usual way by imposing the Leibniz rule
Under the configuration pairing, one checks that this gives the following cooperad structure on . To give an operad structure is to give a map for every tree of the form
where the grafting vertex is the th vertex. Label the two vertices and . To make a cooperad, we need a map . This map sends to , where the edges of and are specified by the following procedure:
For any edge of , consider the leaves and in the tree above. Let be the meet of and (so that or ), and let be the branches of over which and lie (in this case, one of and will be or ). Then add an edge to from to .
The map is an isomorphism of cooperads.
The map is an isomorphism of operads.
Consider the composite
Consider the homotopy where at time , the disks in the first factor are scaled by and the disks in the second factor are scaled by . As , we see that this approaches the projection onto the th factor followed by , as promised.