Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners

This book presents the (to date) most general approach to combinatorial constructions of topological quantum field theories (TQFTs) in three dimensions. The authors describe extended TQFTs as double functors between two naturally defined double categories: one of topological nature, made of 3-manifolds with corners, the other of algebraic nature, made of linear categories, functors, vector spaces and maps. Atiyah's conventional notion of TQFTs as well as the notion of modular functor from axiomatic conformal field theory are unified in this concept. A large class of such extended modular catergory is constructed, assigning a double functor to every abelian modular category, which does not have to be semisimple. TOC:Introduction and summary of results.- The double category of framed, relative 3-cobordisms.- Tangle-categories and presentation of cobordisms.- Isomorphism between tangle and cobordism categories.- Monoidal categories and monoidal 2-categories.- Coends and construction of Hopf algebras.- Construction of TQFT-Double Functors.- Generalization of modular functor.- A: From quantum field theory of axiomatics.- B: Double categories and double functors.- C: Thick tangles.