In many areas of mathematics some "higher operations" are arising. These
havebecome so important that several research projects refer to such
expressions. Higher operationsform new types of algebras. The key to
understanding and comparing them, to creating invariants of their action
is operad theory. This is a point of view that is 40 years old in
algebraic topology, but the new trend is its appearance in several other
areas, such as algebraic geometry, mathematical physics, differential
geometry, and combinatorics. The present volume is the first
comprehensive and systematic approach to algebraic operads. An operad is
an algebraic device that serves to study all kinds of algebras
(associative, commutative, Lie, Poisson, A-infinity, etc.) from a
conceptual point of view. The book presents this topic with an emphasis
on Koszul duality theory. After a modern treatment of Koszul duality for
associative algebras, the theory is extended to operads. Applications to
homotopy algebra are given, for instance the Homotopy Transfer Theorem.
Although the necessary notions of algebra are recalled, readers are
expected to be familiar with elementary homological algebra. Each
chapter ends with a helpful summary and exercises. A full chapter is
devoted to examples, and numerous figures are included.
After a low-level chapter on Algebra, accessible to (advanced)
undergraduate students, the level increases gradually through the book.
However, the authors have done their best to make it suitable for
graduate students: three appendices review the basic results needed in
order to understand the various chapters. Since higher algebra is
becoming essential in several research areas like deformation theory,
algebraic geometry, representation theory, differential geometry,
algebraic combinatorics, and mathematical physics, the book can also be
used as a reference work by researchers.
