Monoidal Categories and Topological Field TheoryThis monograph is devoted to monoidal categories and their connections with 3dimensional 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 Müger 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 TuraevVirotype 3dimensional state sum TQFT from a spherical fusion category. Lastly, in Part 4 this construction is extended to 3manifolds with colored ribbon graphs, yielding a socalled graph TQFT (and, consequently, a 321 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 ReshetikhinTuraev 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.

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Monoidal Categories and Topological Field Theory Vladimir Turaev,Alexis Virelizier No preview available  2017 
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2polyhedron 3manifold ambient isotopy antipode arcs associated bialgebra bijective bimonad Ccolored graph coend coev coevy cokernel colored computed counit coupons cyclic Cset defined definition dim(C dim(i dinatural direct sum edges EndC endpoints epimorphism equality equivalent Example Exercise Figure finite follows forgetful functor Formula functor F fusion kcategory graph TQFT halfbraiding Homc Hopf algebra Hopf monad implies induced Int(M invariant inverse isotopy isotopy invariance klinear homomorphism klinear isomorphism kmodule kernel knotted Cnets left dual Lemma modules monoidal category monoidal functor monoidal product monomorphism morphism f natural isomorphism natural transformation nondegenerate Ob(C obtain pairing Penrose diagram pivotal category positive diagrams Proof Rmatrix regions representative set ribbon graph right duality rigid category Section simple objects skeleton strands strict monoidal strong monoidal functor tensor product Theorem Topological unique morphism vertex vertices weighted diagrams Ye ObſC zero object