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Examples of categorification. (English) Zbl 0895.18004
The purpose of this paper is to consider several simple cases of the inverse relation to the Grothendieck rig construction as constructions of tensor and bitensor categories. The authors consider examples of categorifications and their relevance to the construction of TQFT’s. They restrict their attention to the case of an algebraically closed field \({\mathcal K}\) because this covers the most interesting case of \({\mathcal K}={\mathbb{C}}\) and removes the possibility of having objects in semi-simple categories whose endomorphism algebras are division-algebra extensions of the ground field. One of the examples of categorifications considered in this paper is the question of categorifying the simplest really non-trivial Hopf algebra: the Drinfel’d double of a finite (non-commutative) group algebra.
Reviewer: V.Abramov (Tartu)

MSC:
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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References:
[1] J. Barrett and B. Westbury , ” Spherical Categories ,” (preprint) ( 1992 ). arXiv | MR 1686423 · Zbl 0930.18004
[2] V. Chari and A. Pressley , A Guide to Quantum Groups , Cambridge Univ. Press ( 1994 ). MR 1300632 | Zbl 0839.17009 · Zbl 0839.17009
[3] L. Crane , ” Clock and Category: Is Quantum Gravity Algebraic? ” J. Math. Phys. 36 ( 11 ) ( 1995 ) 6180 - 6193 . MR 1355904 | Zbl 0848.57035 · Zbl 0848.57035
[4] L. Crane and I. Frenkel , ” Four-dimensional Topological Quantum Field Theory, Hopf Categories, and the Canonical Bases ,” J. Math. Phys. 35 ( 10 ) ( 1994 ) 5136 - 5145 . MR 1295461 | Zbl 0892.57014 · Zbl 0892.57014
[5] L. Crane and D.N. Yetter , ” On Algebraic Structures Implicit in Topological Quantum Field Theories ,” J. Knot Th. and Ram. (to appear). Zbl 0935.57025 · Zbl 0935.57025
[6] L. Crane and D.N. Yetter , ” A Categorical Construction of 4D Topological Quantum Field Theories ,” in Quantum Topology (L.H. Kauffman and R.A. Baadhio, eds.) World Scientific ( 1993 ) 120 - 130 . Zbl 0841.57030 · Zbl 0841.57030
[7] L. Crane and D.N. Yetter , ” Deformations of (Bi)tensor Categories ” (in preparation). · Zbl 0916.18005
[8] L. Crane , L.H. Kauffman and D.N. Yetter , ” State-Sum Invariants of 4-Manifolds ,” J. Knot Th. and Ram. (to appear). MR 1452438 | Zbl 0883.57021 · Zbl 0883.57021
[9] P. Deligne , ” Catégories Tannakiennes ,” in The Grothendieck Festschrift , Volume II (P. Cartier, et al. eds.) Birkhäuser ( 1990 ) 111 - 196 . MR 1106898 | Zbl 0727.14010 · Zbl 0727.14010
[10] R. Dijkgraaf , V. Pasquier and P. Roche , ” Quasi-Hopf Algebras, Group Cohomology and Orbifold Models ,” in Integrable Models and Quantum Groups (M. Carfora et al. eds.) World Scientific ( 1992 ). MR 1178543 | Zbl 0925.16024 · Zbl 0925.16024
[11] R. Dijkgraaf and E. Witten , ” Topological Gauge Theories and Group Cohomology ,” Comm. Math. Phys. 129 ( 2 ) ( 1990 ) 393 - 429 . Article | MR 1048699 | Zbl 0703.58011 · Zbl 0703.58011
[12] M.M. Kapranov and V.A. Voevodsky , ” 2-Categories and Zamolodchikov Tetrahedral Equations ,” Proceedings of Symposia in Pure Mathematics 56 ( 2 ) ( 1994 ) 177 - 259 . MR 1278735 | Zbl 0809.18006 · Zbl 0809.18006
[13] R. Gordon , A.J. Power , and R. Street , ” Coherence for Tricategories ” (preprint) ( 1993 ). MR 1261589 | Zbl 0836.18001 · Zbl 0836.18001
[14] S. Mac Lane , Categories for the Working Mathematician , Springer-Verlag ( 1971 ). MR 1712872 | Zbl 0232.18001 · Zbl 0232.18001
[15] S. Majid , ” Quasitriangular Hopf Algebras and Yang-Baxter Equations ,” Int. J. of Mod. Phys. A 5 ( 1 ) ( 1990 ) 1 - 91 . MR 1027945 | Zbl 0709.17009 · Zbl 0709.17009
[16] M. Wakui , ” On Dijkgraaf-Witten Invariant for 3-Manifolds ,” Osaka J. of Math. 29 ( 4 ) ( 1992 ) 675 - 696 . MR 1192735 | Zbl 0786.57008 · Zbl 0786.57008
[17] D.N. Yetter , ” Topological Quantum Field Theories Associated to Finite Groups and Crossed G-Sets ,” J. Knot Th. and Ram. 1 ( 1 ) ( 1992 ) 1 - 20 . MR 1155090 | Zbl 0770.57010 · Zbl 0770.57010
[18] D.N. Yetter , ” Categorical Linear Algebra-A Setting for Questions for Physics and Low-Dimensional Topology ,” (preprint) ( 1993 ).
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