I have a couple of questions concerning basic notions in operad theory.
- What is the ideological difference between symmetric and non-symmetric operads? I think about the difference in the following way. Symmetric operads are collections of boxes (operations) with slots with an additional structure: it is possible to permute the slots. The nonsymmetric case lacks this permutation structure: the order of inputs is fixed. Is this understanding correct?
- Is there any good reference on the constructions of free symmetric/non-symmetric operads? I am looking for a rather informal explanation. Is it true that the first can be understood in terms of labeled rooted trees with vertices decorated by elements of some $\Sigma$-module and the latter in terms of planar trees?
- Can the operads of rooted labeled trees and planar trees be described as free operads generated by some $\Sigma$-modules?
- It seems to me (I hope that I understand the definitions correctly) that the operad of labeled rooted trees is not the symmetrization of the operad of planar trees. Is this correct?
