Frobenius Algebras and 2-D Topological Quantum Field Theories
This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The precise formulation of the theorem and its proof is given in terms of monoidal categories, and the main purpose of the book is to develop these concepts from an elementary level, and more generally serve as an introduction to categorical viewpoints in mathematics. Rather than just proving the theorem, it is shown how the result fits into a more general pattern concerning universal monoidal categories for algebraic structures. Throughout, the emphasis is on the interplay between algebra and topology, with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. The book will prove valuable to students or researchers entering this field who will learn a host of modern techniques that will prove useful for future work.
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1-manifold 2Cob A-linear associativity axioms bilinear called canonical cartesian product characterised coalgebra cobordism classes commutative Frobenius algebras commutative Frobenius object comonoid compatible composition comultiplication connected components consider construction copairing corresponding counit cylinder defined definition denote diagrams commute diffeomorphism disjoint union dots dual empty equation equivalence of categories equivalent example Exercise finite dimension finite sets finite-dimensional FinSet free monoidal Frobenius algebra Frobenius form Frobenius relation Frobenius structure given graphical graphs identity arrows identity map in-boundary invertible isomorphism k-algebra Lemma linear map manifold with boundary monoid homomorphism monoidal structure multiplication map nCob neutral element nondegenerate nontrivial normal form Note notion oriented cobordisms out-boundary pairing ft permutation phism precisely proof quantum field theories right A-module ring sense Show smooth structure strands surfaces symmetric group symmetric monoidal category symmetric monoidal functor tensor product theorem topological quantum field TQFT twist map unique universal property Vect Vectk vector space
Page 234 - BAEZ and JAMES DOLAN. Higher-dimensional algebra and topological quantum field theory. J. Math. Phys. 36 (1995), 6073-6105 (q-alg/9503002).