
Chapter 5
Symmetric Operads and Shuffle
Operads
5.1 Introduction
This chapter continues developing methods for handling operations with
several arguments. In Chapter 3, we used nonsymmetric operads, which al-
lowed us to deal with substitutions of operations. In that chapter, we remarked
that the language of nonsymmetric operads is rich enough to express proper-
ties like associativity, where all arguments appear in the same order, but is
insufficient for the properties like the Jacobi identity in Lie algebras. If one
attempts to combine substitutions of operations with permutations of argu-
ments, the natural notion to deal with is that of a symmetric op ...