A Note on Strict Commutativity of a Monoidal Product

Youngsoo Kim

doi:doi:10.11648/j.pamj.20160505.13

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A Note on Strict Commutativity of a Monoidal Product

Youngsoo Kim

Published in Pure and Applied Mathematics Journal (Volume 5, Issue 5)

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

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[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.

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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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TY  - JOUR
T1  - A Note on Strict Commutativity of a Monoidal Product
AU  - Youngsoo Kim
Y1  - 2016/09/21
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160505.13
DO  - 10.11648/j.pamj.20160505.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 155
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PB  - Science Publishing Group
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AB  - 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.
VL  - 5
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Author Information

Youngsoo Kim

Department of Mathematics, Tuskegee University, Tuskegee, USA

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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

Copy | Download

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

Copy | Download

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - A Note on Strict Commutativity of a Monoidal Product
AU  - Youngsoo Kim
Y1  - 2016/09/21
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160505.13
DO  - 10.11648/j.pamj.20160505.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 155
EP  - 159
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160505.13
AB  - 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.
VL  - 5
IS  - 5
ER  -

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