A Note on Strict Commutativity of a Monoidal Product
Pure and Applied Mathematics Journal
Volume 5, Issue 5, October 2016, Pages: 155-159
Received: Aug. 25, 2016; Accepted: Sep. 5, 2016; Published: Sep. 21, 2016
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Youngsoo Kim, Department of Mathematics, Tuskegee University, Tuskegee, USA
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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.
Symmetric Monoidal Category, Strict Commutativity, Monoidal Product
To cite this article
Youngsoo Kim, A Note on Strict Commutativity of a Monoidal Product, Pure and Applied Mathematics Journal. Vol. 5, No. 5, 2016, pp. 155-159. doi: 10.11648/j.pamj.20160505.13
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This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
C. Balteanu, Z. Fiedorowicz, R. Schwänzl, and R. Vogt, Iterated monoidal categories, Adv. Math. 176 (2003), no. 2, 277–349. MR 1982884.
Mitya Boyarchenko, Associativity constraints in monoidal categories, http://math.uchicago.edu/~may/TQFT/Boyarchenko%20on%20associativity.pdf.
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.
Bertrand J. Guillou, Strictification of categories weakly enriched in symmetric monoidal categories, Theory Appl. Categ. 24 (2010), No. 20, 564–579. MR 2770075.
Nick Gurski and Ang ́elica M. Osorno, Infinite loop spaces, and coherence for symmetric monoidal bicategories, Adv. Math. 246 (2013), 1–32. MR 3091798.
André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. MR 1250465.
Saunders Mac Lane, Natural associativity and commutativity, Rice Univ. Studies 49 (1963), no. 4, 28–46. MR 0170925.
Saunders Mac Lane, Categories for the working mathematician, second ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872.
Peter Schauenburg, Turning monoidal categories into strict ones, New York J. Math. 7 (2001), 257–265 (electronic). MR 1870871 (2003d: 18013).
Mirjam Solberg, Weak braided monoidal categories and their homotopy colimits, Theory Appl. Categ. 30 (2015), 40–48. MR 3306878.
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