Etingof Pavel et al. Tensor Categories

American Mathematical Society, 2015. — 362 p.Tensor categories should be thought of as counterparts of rings in the world of categories. They are ubiquitous in noncommutative algebra and representation theory, and also play an important role in many other areas of mathematics, such as algebraic geometry, algebraic topology, number theory, the theory of operator algebras, mathematical physics, and theoretical computer science (quantum computation).The definition of a monoidal category first appeared in 1963 in the work of Mac Lane [Mac1], and later in his classical book [Mac2] (first published in 1971).
Mac Lane proved two important general theorems about monoidal categories – the coherence theorem and the strictness theorem, and also defined symmetric and braided monoidal categories. Later, Saavedra-Rivano in his thesis under the direction of Grothendieck [Sa], motivated by the needs of algebraic geometry and number theory (more specifically, the theory of motives), developed a theory of Tannakian categories, which studies symmetric monoidal structures on abelian categories (the prototypical example being the category of representations of an algebraic group). This theory was simplified and further developed by Deligne and Milne in their classical paper [DelM]. Shortly afterwards, the theory of tensor categories (i.e., monoidal abelian categories) became a vibrant subject, with spectacular connections to representation theory, quantum groups, infinite dimensional Lie algebras, conformal field theory and vertex algebras, operator algebras, invariants of knots and 3-manifolds, number theory, etc., which arose from the works of Drinfeld, Moore and Seiberg, Kazhdan and Lusztig, Jones, Witten, Reshetikhin and Turaev, and many others. Initially, in many of these works tensor categories were merely a tool for solving various concrete problems, but gradually a general theory of tensor categories started to emerge, and by now there are many deep results about properties and classification of tensor categories, and the theory of tensor categories has become fairly systematic. The goal of this book is to provide an accessible introduction to this theory.