Progress in Mathematics
2 primary works
Book 208
From the reviews: "This is an excellent exposition about abelian Reidemeister torsions for three-manifolds." -Zentralblatt Math
"This monograph contains a wealth of information many topologists will find very handy. ...Many of the new points of view pioneered by Turaev are gradually becoming mainstream and are spreading beyond the pure topology world. This monograph is a timely and very useful addition to the scientific literature." -Mathematical Reviews
Book 322
Monoidal Categories and Topological Field Theory
by Vladimir Turaev and Alexis Virelizier
This monograph is devoted to monoidal categories and their connections with 3-dimensional topological field theories. Starting with basic definitions, it proceeds to the forefront of current research.
Part 1 introduces monoidal categories and several of their classes, including rigid, pivotal, spherical, fusion, braided, and modular categories. It then presents deep theorems of Muger on the center of a pivotal fusion category. These theorems are proved in Part 2 using the theory of Hopf monads. In Part 3 the authors define the notion of a topological quantum field theory (TQFT) and construct a Turaev-Viro-type 3-dimensional state sum TQFT from a spherical fusion category. Lastly, in Part 4 this construction is extended to 3-manifolds with colored ribbon graphs, yielding a so-called graph TQFT (and, consequently, a 3-2-1 extended TQFT). The authors then prove the main result of the monograph: the state sum graph TQFT derived from any spherical fusion category is isomorphic to the Reshetikhin-Turaev surgery graph TQFT derived from the center of that category.The book is of interest to researchers and students studying topological field theory, monoidal categories, Hopf algebras and Hopf monads.