Traces in monoidal categories
Authors:
Stephan Stolz and Peter Teichner
Journal:
Trans. Amer. Math. Soc. 364 (2012), 44254464
MSC (2010):
Primary 18D10; Secondary 46A32, 81T99
Published electronically:
March 29, 2012
MathSciNet review:
2912459
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Abstract: This paper contains the construction, examples and properties of a trace and a trace pairing for certain morphisms in a monoidal category with switching isomorphisms. Our construction of the categorical trace is a common generalization of the trace for endomorphisms of dualizable objects in a balanced monoidal category and the trace of nuclear operators on a topological vector space with the approximation property. In a forthcoming paper, applications to the partition function of supersymmetric field theories will be given.
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 P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309317 MR 0402468 (53:6288)
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 A. Joyal and R. Street, The geometry of tensor calculus I, Adv. Math. 88 (1991), 55112. MR 1113284 (92d:18011)
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 A. Joyal and R. Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 2078. MR 1250465 (94m:18008)
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 A. Joyal, R. Street and D. Verity, Traced monoidal categories, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 447468. MR 1357057 (96m:18014)
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 G. Kelly, Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series, 64. Cambridge University Press, CambridgeNew York, 1982. MR 651714 (84e:18001)
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 G. Köthe, Topological vector spaces. II., Grundlehren der Mathematischen Wissenschaften, 237. SpringerVerlag, New YorkBerlin, 1979. xii+331 pp. MR 551623 (81g:46001)
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 H. H. Schaefer and M. P. Wolff, Topological vector spaces, Second edition. Graduate Texts in Mathematics, 3. SpringerVerlag, New York, 1999. xii+346 pp.MR 1741419 (2000j:46001)
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 G. Segal, The definition of conformal field theory. Topology, geometry and quantum field theory, 423  577, London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, 2004. MR 2079383 (2005h:81334)
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 S. Stolz and P. Teichner, Super symmetric field theories and integral modular forms, preprint available at http://web.me.com/teichner/Math/Surveys.html
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 S. Stolz and P. Teichner, Super symmetric Euclidean field theories and generalized cohomology, preprint available at the same website.
 [Sz]
 A. Szankowski, does not have the approximation property, Acta Math. 147 (1981), no. 12, 89108. MR 631090 (83a:46033)
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Additional Information
Stephan Stolz
Affiliation:
Department of Mathematics, University of Notre Dame, South Bend, Indiana 46556
Peter Teichner
Affiliation:
Max Planck Institute for Mathematics, PO Box 7280, 53072 Bonn, Germany
DOI:
http://dx.doi.org/10.1090/S000299472012056157
Received by editor(s):
October 21, 2010
Received by editor(s) in revised form:
April 29, 2011
Published electronically:
March 29, 2012
Additional Notes:
Both authors were partially supported by NSF grants. They would like to thank the referee for many valuable suggestions. The first author visited the second author at the MaxPlanckInstitut in Bonn during the Fall of 2009 and in July 2010. He would like to thank the institute for its support and for its stimulating atmosphere.
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© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
