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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Traces in monoidal categories
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by Stephan Stolz and Peter Teichner PDF
Trans. Amer. Math. Soc. 364 (2012), 4425-4464 Request permission

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 super-symmetric field theories will be given.
References
  • Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar, IV, Lecture Notes in Mathematics, Vol. 137, Springer, Berlin, 1970, pp. 1–38. MR 0272852
  • Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 402468, DOI 10.1007/BF02392270
  • André Joyal and Ross Street, The geometry of tensor calculus. I, Adv. Math. 88 (1991), no. 1, 55–112. MR 1113284, DOI 10.1016/0001-8708(91)90003-P
  • André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. MR 1250465, DOI 10.1006/aima.1993.1055
  • André Joyal, Ross Street, and Dominic Verity, Traced monoidal categories, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 447–468. MR 1357057, DOI 10.1017/S0305004100074338
  • Gregory Maxwell Kelly, Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series, vol. 64, Cambridge University Press, Cambridge-New York, 1982. MR 651714
  • Gottfried Köthe, Topological vector spaces. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 237, Springer-Verlag, New York-Berlin, 1979. MR 551623
  • G. L. Litvinov, Approximation properties of locally convex spaces and the problem of uniqueness of the trace of a linear operator [translation of Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. No. 39 (1983), 73–87; MR0734687 (85g:47028)], Selecta Math. Soviet. 11 (1992), no. 1, 25–40. Selected translations. MR 1155897
  • Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
  • Ralf Meyer, Local and analytic cyclic homology, EMS Tracts in Mathematics, vol. 3, European Mathematical Society (EMS), Zürich, 2007. MR 2337277, DOI 10.4171/039
  • H. H. Schaefer and M. P. Wolff, Topological vector spaces, 2nd ed., Graduate Texts in Mathematics, vol. 3, Springer-Verlag, New York, 1999. MR 1741419, DOI 10.1007/978-1-4612-1468-7
  • Graeme Segal, The definition of conformal field theory, Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, Cambridge, 2004, pp. 421–577. MR 2079383
  • S. Stolz and P. Teichner, Super symmetric field theories and integral modular forms, preprint available at http://web.me.com/teichner/Math/Surveys.html
  • S. Stolz and P. Teichner, Super symmetric Euclidean field theories and generalized cohomology, preprint available at the same website.
  • Andrzej Szankowski, $B({\cal H})$ does not have the approximation property, Acta Math. 147 (1981), no. 1-2, 89–108. MR 631090, DOI 10.1007/BF02392870
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Additional Information
  • Stephan Stolz
  • Affiliation: Department of Mathematics, University of Notre Dame, South Bend, Indiana 46556
  • MR Author ID: 167655
  • Peter Teichner
  • Affiliation: Max Planck Institute for Mathematics, PO Box 7280, 53072 Bonn, Germany
  • 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 Max-Planck-Institut 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.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 4425-4464
  • MSC (2010): Primary 18D10; Secondary 46A32, 81T99
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05615-7
  • MathSciNet review: 2912459