Monoidal categories and topological field theory
type: Item
instance of: scholarly article
title: Monoidal categories and topological field theory
zbMATH Open document ID: 1423.18001
DOI: 10.1007/978-3-319-49834-8
author: Alexis Virelizier
author: Vladimir Turaev
published in: Progress in Mathematics
publication date: 2017-06-30T00:00:00Z
review text: The study of monoidal categories was initiated in [\textit{S. MacLane}, Rice Univ. Stud. 49, No. 4, 28--46 (1963; Zbl 0244.18008)] and [\textit{J. Benabou}, C. R. Acad. Sci., Paris 256, 1887--1890 (1963; Zbl 0111.02201)]. The intimate relationship between monoidal categories and topology was sparked by \textit{R. Penrose} [Proc. Conf. Math. Inst. 1969, 221--244 (1971; Zbl 0216.43502)]. Morphisms in monoidal categories and operations on them may be represented by pictures, which leads to a fantastic graphical calculus allowing to replace lengthy algebraic computations by simple topological arguments. Monoidal categories play a significant role in quantum topology, an arena of mathematics and theoretical physics founded by Vaughan Jones and Edward Witten in the 1980s, which studies Topological Quantum Field Theory (TQFT) as well as related invariants such as knots, links manifolds, etc. This monograph is engaged in connections between monoidal categories and $3$-dimensional TQFTs. The first named author of the monograph introduced two fundamental constructions of $3$-dimensional TQFTs in collaboration with Nikolai Reshetikhin and Oleg Viro in the late 1980s, the Reshetikhin-Turaev construction [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007)] making use of surgery on $3$-manifolds to derive a TQFT from a modular category while the Turaev-Viro construction [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009)] exploiting state sums on on skeletons of $3$-manifolds to derive a TQFT from a spherical fusion category. The latter construction originally involved representations of quantum groups, whose categorical formulation was given in [\textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)]. In 1995 the first named author of the monograph conjectured that the state sum TQFT derived from a spherical f…
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 18-02
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 81T45
zbMATH DE Number: 6738086
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-49834-8
OpenAlex ID: W2728096305
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description: scientific article