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A Note on Strict Commutativity of a Monoidal Product

Received: 25 August 2016     Accepted: 5 September 2016     Published: 21 September 2016
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Abstract

It is well known that a monoidal category is (monoidally) equivalent to a strict monoidal category that is a monoidal category with a strictly associative product. In this article, we discuss strict commutativity and prove a necessary and sufficient condition for a symmetric monoidal category to be equivalent to another symmetric monoidal category with a strictly commutative monoidal product.

Published in Pure and Applied Mathematics Journal (Volume 5, Issue 5)
DOI 10.11648/j.pamj.20160505.13
Page(s) 155-159
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Symmetric Monoidal Category, Strict Commutativity, Monoidal Product

References
[1] C. Balteanu, Z. Fiedorowicz, R. Schwänzl, and R. Vogt, Iterated monoidal categories, Adv. Math. 176 (2003), no. 2, 277–349. MR 1982884.
[2] Mitya Boyarchenko, Associativity constraints in monoidal categories, http://math.uchicago.edu/~may/TQFT/Boyarchenko%20on%20associativity.pdf.
[3] V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283.
[4] Bertrand J. Guillou, Strictification of categories weakly enriched in symmetric monoidal categories, Theory Appl. Categ. 24 (2010), No. 20, 564–579. MR 2770075.
[5] Nick Gurski and Ang ́elica M. Osorno, Infinite loop spaces, and coherence for symmetric monoidal bicategories, Adv. Math. 246 (2013), 1–32. MR 3091798.
[6] André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. MR 1250465.
[7] Saunders Mac Lane, Natural associativity and commutativity, Rice Univ. Studies 49 (1963), no. 4, 28–46. MR 0170925.
[8] Saunders Mac Lane, Categories for the working mathematician, second ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872.
[9] Peter Schauenburg, Turning monoidal categories into strict ones, New York J. Math. 7 (2001), 257–265 (electronic). MR 1870871 (2003d: 18013).
[10] Mirjam Solberg, Weak braided monoidal categories and their homotopy colimits, Theory Appl. Categ. 30 (2015), 40–48. MR 3306878.
Cite This Article
  • APA Style

    Youngsoo Kim. (2016). A Note on Strict Commutativity of a Monoidal Product. Pure and Applied Mathematics Journal, 5(5), 155-159. https://doi.org/10.11648/j.pamj.20160505.13

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    ACS Style

    Youngsoo Kim. A Note on Strict Commutativity of a Monoidal Product. Pure Appl. Math. J. 2016, 5(5), 155-159. doi: 10.11648/j.pamj.20160505.13

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    AMA Style

    Youngsoo Kim. A Note on Strict Commutativity of a Monoidal Product. Pure Appl Math J. 2016;5(5):155-159. doi: 10.11648/j.pamj.20160505.13

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  • @article{10.11648/j.pamj.20160505.13,
      author = {Youngsoo Kim},
      title = {A Note on Strict Commutativity of a Monoidal Product},
      journal = {Pure and Applied Mathematics Journal},
      volume = {5},
      number = {5},
      pages = {155-159},
      doi = {10.11648/j.pamj.20160505.13},
      url = {https://doi.org/10.11648/j.pamj.20160505.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160505.13},
      abstract = {It is well known that a monoidal category is (monoidally) equivalent to a strict monoidal category that is a monoidal category with a strictly associative product. In this article, we discuss strict commutativity and prove a necessary and sufficient condition for a symmetric monoidal category to be equivalent to another symmetric monoidal category with a strictly commutative monoidal product.},
     year = {2016}
    }
    

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Author Information
  • Department of Mathematics, Tuskegee University, Tuskegee, USA

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