In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad , one defines an algebra over to be a set together with concrete operations on this set which behave just like the abstract operations of . For instance, there is a Lie operad such that the algebras over are precisely the Lie algebras; in a sense abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations. History Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1969 and by J. Peter May in 1970. The word "operad" was created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer). Interest in operads was considerably renewed in the early 90s when, based on early insights of Maxim Kontsevich, Victor Ginzburg and Mikhail Kapranov discovered that some duality phenomena in rational homotopy theory could be explained using Koszul duality of operads. Operads have since found many applications, such as in deformation quantization of Poisson manifolds, the Deligne conjecture, or graph homology in the work of Maxim Kontsevich and Thomas Willwacher. Intuition Suppose is a set and for we define , the set of all functions from the cartesian product of copies of to . We can compose these functions: given , , the function is defined as follows: given arguments from , we divide them into blocks, the first one having arguments, the second one arguments, etc., and then apply to the first block, to the second block, etc. We then apply to the list of values obtained from in such a way. We can also permute arguments, i.e. we have a right action of the symmetric group on , defined by for , and . The definition of a symmetric operad given below captures the essential properties of these two operations and . Definition Non-symmetric operad A non-symmetric operad (sometimes called an operad without permutations, or a non- or plain operad) consists of the following: a sequence of sets, whose elements are called -ary operations, an element in called the identity, for all positive integers ,