ABSTRACT
Symmetric operads were invented by Peter May for purely topological reasons (to study spectra); see [193, 242] for some historical (and even prehistorical) background. (It is worth remarking that, though invented and, until 1990s, almost exclusively used by topologists, operads could have been rediscovered by experts in combinatorics in the context of combinatorial species [142], as monoids in species with respect to the partitional composition [94, 141].) Notably, similar to the phenomenon we observed for twisted associative algebras in Chapter 4, the presence of symmetries makes it more difficult to have a working formalism of normal forms. The solution to this problem is similar to the one we exhibited for twisted associative algebras; we will explain how to ignore the symmetries for most purposes by dealing with shuffle operads instead of symmetric operads.
