Operad theory

In mathematics, an operad is concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. Operads generalize the various associativity properties already observed in algebras and coalgebras such as Lie algebras or Poisson algebras by modeling computational trees within the algebra. Algebras are to operads as group representations are to groups. An operad can be seen as a set of operations, each one having a fixed finite number of inputs (arguments) and one output, which can be composed one with others. They form a category-theoretic analog of universal algebra. In mathematics, an operad is concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. Operads generalize the various associativity properties already observed in algebras and coalgebras such as Lie algebras or Poisson algebras by modeling computational trees within the algebra. Algebras are to operads as group representations are to groups. An operad can be seen as a set of operations, each one having a fixed finite number of inputs (arguments) and one output, which can be composed one with others. They form a category-theoretic analog of universal algebra. Operads originate in algebraic topology from the study of iterated loop spaces by J. Michael Boardman and Rainer M. Vogt, and J. Peter May. 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. A non-symmetric operad (sometimes called an operad without permutations, or a non- Σ {displaystyle Sigma } or plain operad) consists of the following: