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S‐Structures for k ‐Linear Categories and the Definition of a Modular Functor
Author(s) -
Tillmann Ulrike
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798006383
Subject(s) - functor , mathematics , axiom , hopf algebra , pure mathematics , natural transformation , enriched category , adjoint functors , algebraic structure , categorical variable , field (mathematics) , equivalence of categories , algebra over a field , category theory , algebraic theory , quantum field theory , topological quantum field theory , monoidal category , algebraic number , geometry , mathematical analysis , statistics , mathematical physics
Ideas from string theory and quantum field theory have been the motivation for new invariants of knots and 3‐dimensional manifolds which have been constructed from complex algebraic structures such as Hopf algebras [ 17 , 22 ], monoidal categories with additional structure [ 24 ], and modular functors [ 14 , 23 ]. These constructions are closely related. Here we take a unifying categorical approach based on a natural 2‐dimensional generalisation of a topological field theory in the sense of Atiyah [ 1 ], and show that the axioms defining these complex algebraic structures are a consequence of the underlying geometry of surfaces.