# An exceptional review of Tensor Categories

I have fallen behind in my posts about exceptional reviews in MathSciNet.  Let’s try to catch up with an excellent review of the book Tensor Categories by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik.  The review is written by Julien Bichon, who has written (so far) over 60 reviews.  There are several very good aspects of Bichon’s review.  One can immediately see that it is thorough.  Beyond that, Bichon has organized the review nicely, telling the reader first what the book is about (and also what it is not about) and why it might be interesting.  He also explains what the expectations on the reader are.  Bichon then provides a chapter-by-chapter discussion of the contents, calling out the highlights and pointing out where helpful examples occur.  The review finishes with a summary and a recommendation.

MR3242743
Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor
Tensor categories.
Mathematical Surveys and Monographs, 205.
American Mathematical Society, Providence, RI, 2015. xvi+343 pp.
ISBN: 978-1-4704-2024-6
18D10 (16T05)

This book provides a rather complete treatment of the theory of tensor categories, starting from the very basic facts and finishing with the most recent developments. The focus is on tensor categories themselves, viewed as generalizations of (representations of) groups, Lie algebras and Hopf algebras, not on their many connections with other fields of mathematics such as low-dimensional topology, operator algebras, quantum mechanics and field theory, theory of motives, etc. One of the main aims is to explain how classical results on Hopf algebras can be derived from more general ones on tensor categories and how Hopf algebras are better understood through the prism of tensor categories.

This monograph, written by the main actors in the field, is the first one entirely devoted to tensor categories (previous books or parts of books on the subject were more focused on applications, e.g. to low-dimensional topology), and covers many striking developments of recent years which had not yet appeared in textbook form. It is thus a very welcome and useful addition to the literature.

The main prerequisite for reading this book is some knowledge of abelian categories and representation theory of groups. Some Hopf algebra background is welcome as well, but does not seem to be truly necessary, and beginning Hopf algebra theory from the categorical perspective can be, in the reviewer’s opinion, a good way to start.

The book consists of 343 pages and nine chapters, each one finishing with a “bibliographical notes” section and, starting with Chapter 5, an “other results” section. The precise contents are as follows.

Chapter 1 is a review of the theory of abelian categories, presented in a form that will be suitable for the rest of the book. The chapter emphasizes the role of coalgebras, discusses locally finite categories, coalgebras, the coend construction and reconstruction of coalgebras, and contains in particular a sketch of a proof of an important result of Takeuchi: any essentially small locally finite abelian $k$-linear ($k$ is a field) category is equivalent to the category of comodules over a (pointed) coalgebra (unfortunately, the sketch of the proof of the crucial Theorem 1.10.1 uses Ind-objects, not defined in the text, as well as an undefined external tensor product, so the reader who does not already know a proof will have to go to external references to really understand it). Deligne’s tensor product of locally finite abelian categories is constructed, using the previous result of Takeuchi.

Chapter 2 provides the most general facts and results on monoidal categories. A pleasant and original feature is the slightly unusual definition used, which “allows one not to worry too much about the units”. Short proofs of the Mac Lane coherence and strictness theorems are given, rigid monoidal categories are discussed, and a number of basic examples are given.

Chapter 3, a technical chapter needed later in the book, collects useful results on $\Bbb Z_+$-rings, which serve as Grothendieck rings of tensor categories, and develops in particular the crucial theory of Frobenius-Perron dimension.

Chapter 4 deals with the general theory of (multi-)tensor categories, proving some basic results such as exactness of the tensor product and semisimplicity of the tensor unit, and introducing key notions and constructions such as the Frobenius-Perron dimension of objects, pivotal and spherical categories, the adjoint subcategory and universal grading, and equivariantization.

In Chapter 5, the authors introduce Hopf algebras in the framework of Tannaka-Kreĭn reconstruction theory: they explain that their categories of representations (or corepresentations) are tensor categories and correspond, conversely, to tensor categories equipped with a fibre functor. Several examples are presented, and the structure theorem of cocommutative Hopf algebras over an algebraically closed field of characteristic zero is proved in the light of tensor categories. Pointed Hopf algebras, pointed tensor categories, twists and quasi-Hopf algebras are discussed as well.

Chapter 6 is devoted to finite tensor categories. Projective objects are studied and are shown to be injective. Finite-dimensional quasi-Hopf algebras, whose representation categories are characterized as finite tensor categories with integral Frobenius-Perron dimension for all objects, are shown to be Frobenius algebras, and their theory of integrals is discussed. A categorical generalization of the Nichols-Zoeller theorem (a finite-dimensional Hopf algebra is free over its Hopf subalgebra; this is the Hopf algebra generalization of the Lagrange theorem) is proved.

In the study of tensor categories, module categories play a role that is similar to that of modules in the study of rings. This theory is presented in Chapter 7. The authors first develop the basic theory of module categories over monoidal categories, then turn to the abelian setting, introducing the notion of an exact module category over a finite tensor category (somewhat analogous to the notion of a projective module over a ring). They show that module categories arise as categories of modules over algebras in tensor categories, making algebras in tensor categories the main technical tool for the study of module categories. They study the category of module functors between two module categories, with the Drinfeld center construction as an important special case. For Hopf algebras, this gives rise to the famous Drinfeld double construction and Yetter-Drinfeld modules. Next, the authors discuss dual categories and categorical Morita equivalence of tensor categories, prove the Fundamental Theorem for Hopf modules and bimodules over a Hopf algebra, and prove the categorical version of Radford’s $S^4$ formula. They develop as well the theory of categorical dimensions of fusion categories. The chapter ends with a discussion of weak Hopf algebras, which are generalizations of Hopf algebras arising from semisimple module categories via reconstruction theory.

Chapter 8 is devoted to the theory of braided categories, i.e. tensor categories for which the tensor product is commutative in an appropriate sense. This is an important part of the theory of tensor categories because of, among other things, its connections to low-dimensional topology. The authors discuss pointed braided categories (corresponding to quadratic forms on abelian groups) and quasitriangular Hopf algebras (arising from braided categories through reconstruction theory) and show that the center of a tensor category is a braided category. They develop a theory of commutative algebras in braided categories and show that modules over such an algebra form a tensor category. They study the theory of factorisable, ribbon and modular categories, the $S$-matrix, and Gauss sums, and prove the Verlinde formula and the existence of an ${\rm SL}_2(\Bbb Z)$-action, as well as the Anderson-Moore-Vafa theorem (the central charge and twists of a modular category are roots of unity). The chapter also contains the theory of centralizers and projective centralizers in braided categories, de-equivariantization of braided categories, and braided $G$-crossed categories.

In Chapter 9, the authors mostly discuss results about fusion categories, i.e. finite tensor categories that are semisimple, using the various tools constructed in the previous chapters. They prove Ocneanu rigidity, that is, the statement that fusion categories in characteristic zero have no deformations. This implies in particular that a fusion category only has a finite number of fibre functors, as well as Stefan’s theorem that the number of isomorphism classes of finite-dimensional semisimple Hopf algebras of a given dimension is finite. They develop as well the theory of dual categories and pseudo-unitary categories (showing that they have a canonical spherical structure, the categorical analog of the statement that a semisimple Hopf algebra in characteristic zero is involutive) and study integral, weakly integral, group-theoretical, and weakly group-theoretical fusion categories. They discuss symmetric and Tannakian fusion categories, providing a proof of Deligne’s theorem on the classification of such categories, and discuss its nonfusion generalization stating that a symmetric tensor category of subexponential growth is the representation category of a supergroup (with a parity condition). They also give examples of symmetric categories with faster growth, for example Deligne’s categories ${\rm Rep}(S_t)$. Next they provide a criterion of group-theoreticity of a fusion category and show that any integral fusion category of prime power dimension is group theoretical. For categories of dimension $p$ and $p^2$, this gives an explicit classification. They introduce the notion of a solvable fusion category and prove a categorical analog of Burnside’s theorem, stating that a fusion category of dimension $p^aq^b$, where $p, q$ are primes, is solvable. The chapter ends with a discussion of lifting theory for fusion categories (from characteristic $p$ to characteristic zero).

This monograph will be very useful to researchers in tensor categories or connected fields, as well as to beginners, giving many recent results that had not appeared in textbook form. The authors have chosen to develop most results in the greatest generality, which of course is very convenient for researchers wishing to use the results in various situations, and a lot of vocabulary has to be learnt (multi-ring categories, ring-categories, multi-tensor categories, etc.). However, the book is written in a friendly style and is pleasant to read, providing in general enough definitions, motivation and details, and leaving a good amount of exercises to the reader.

In conclusion, this book is certainly a must-have for any researcher working in tensor categories or a related field, and is already the standard reference in the subject.

Reviewed by Julien Bichon