## Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

## Description

An important feature of category theory is that it allows one to imitate, at a higher level, many classical notions of elementary algebra. "Higher level" means that one allows maps (morphisms) not only between algebraic structures, but also between their elements. Such imitation is called categorification. The very notion of a category is a categorification, in this sense, of the notion of a set. Further, monoidal categories categorify monoids, and tensor categories (i.e., monoidal categories with a compatible additive structure), categorify associative rings.

Tensor categories are ubiquitous in mathematics. They arise in representation theory (representation categories of classical and quantum groups), algebraic geometry (categories of coherent sheaves on algebraic varieties, categories of local systems, categories of motives), topology (topological quantum field theory, invariants of knots, links, and 3-manifolds) the theory of operator algebras (bimodule categories for subfactors), 2-dimensional conformal field theory (fusion categories of modules over a vertex operator algebra), quantum statistical mechanics (nonabelian anyons in the quantum Hall effect), etc.

This course will be an attempt to give a detailed introduction to the theory of tensor categories and to review some of its connections to other subjects, time permitting (with a focus on representation-theoretic applications). In particular, we will discuss categorifications of such notions from ring theory as: module, morphism of modules, Morita equivalence of rings, commutative ring, the center of a ring, the centralizer of a subring, the double centralizer property, graded ring, etc.

## Topics

We will develop from scratch the theory of monoidal and tensor categories and module categories, covering the following topics:

- Monoidal categories: the definition, MacLane's coherence theorem, monoidal functors and their natural transformations, equivalence of monoidal categories, rigid monoidal categories, pivotal and spherical categories, dimensions, examples (Hopf algebras/quantum groups).
- Tensor categories: Grothendieck rings, fiber functors and reconstruction theory for Hopf algebras.
- Tensor categories with finitely many simple objects, Frobenius-Perron dimensions, fusion categories, examples.
- Module categories: the definition, examples, properties, algebras in categories, categories of module functors, dual categories. Morita equivalence of tensor categories.

## Prerequisites

Before beginning this course, you are expected to know basic category theory, basic algebra, and fundamentals of group representations. Hopf algebras will often appear in the course, but no previous knowledge of them is expected.

## Assignments

There will be five homework problem sets assigned during the course. It is ok to collaborate on homework if you creatively participate in solving it and understand what you write.

## Calendar

Course calendar. SES # | TOPICS |

1 | Basics of monoidal categories |

2 | Monoidal functors MacLane's strictness theorem |

3 | MacLane coherence theorem Rigid monoidal categories Invertible objects Tensor and multitensor categories |

4 | Tensor product and tensor functors Unit object Grothendieck rings Groupoids Finite abelian categories Fiber functors Coalgebras |

5 | Bialgebras and Hopf algebras |

6 | Quantum groups Skew-primitive elements Pointed tensor categories Coradical filtration Chevalley's theorem and Chevalley property |

7 | Andruskeiwitsch-Schneider conjecture Cartier-Kostant theorem Quasi-bialgebras and quasi-Hopf algebras |

8 | Quantum traces Pivotal categories and dimensions Spherical categories Multitensor cateogries Multifusion rings Frobenius-Perron theorem |

9 | Tensor categories Deligne's tensor product Finite (multi)tensor categories Categorical freeness |

10 | Distinguished invertible object Integrals in quasi-Hopf algebras Cartan matrix Basics of Module categories |

11 | Exact module categories Algebras in categories Internal Hom |

12 | Main Theorem Categories of module functors Dual categories |