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Elementary remarks on units in monoidal categories

Published online by Cambridge University Press:  01 January 2008

JOACHIM KOCK
JOACHIM KOCK*
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain. e-mail: [email protected]
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Abstract

We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellable idempotent (in a 1-categorical sense). This notion is more economical than the usual notion in terms of left-right constraints, and is motivated by higher category theory. To start, we describe the semi-monoidal category of all possible unit structures on a given semi-monoidal category and observe that it is contractible (if non-empty). Then we prove that the two notions of units are equivalent in a strong functorial sense. Next, it is shown that the unit compatibility condition for a (strong) monoidal functor is precisely the condition for the functor to lift to the categories of units, and it is explained how the notion of Saavedra unit naturally leads to the equivalent non-algebraic notion of fair monoidal category, where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered. The paper is self-contained. All arguments are elementary, some of them of a certain beauty.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 1 , January 2008 , pp. 53 - 76
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Bénabou, J.. Catégories avec multiplication. C. R. Acad. Sci. Pari 256 (1963), 1887–1890.Google Scholar
[2]Bénabou, J.. Algèbre élémentaire dans les catégories avec multiplication. C. R. Acad. Sci. Pari 258 (1964), 771–774.Google Scholar
[3]Bénabou, J.. Introduction to bicategories. In Reports of the Midwest Category Seminar, pp. 1–77. Lecture Notes in Math. 47 (Springer-Verlag, 1967).CrossRefGoogle Scholar
[4]Joyal, A. and Kock, J.. Coherence for weak units. Manuscript in preparation.CrossRefGoogle Scholar
[5]Joyal, A. and Kock, J.. Weak units and homotopy 3-types. In Street Festschriff: Categories in Algebra, Geometry and Mathematical Physics, pp. 257–276. Contemp. Math 431 (2007). ArXiv:math.CT/0602084.Google Scholar
[6]Kelly, G. M.. On Mac Lane's conditions for coherence of natural associativities, commutativities, etc. J. Algebr 1 (1964), 397–402.CrossRefGoogle Scholar
[7]Kock, J.. Weak identity arrows in higher categories. Intern. Math. Res. Paper 2006 (2006), 1–54. (ArXiv:math.CT/0507116).Google Scholar
[8]Lane, S. Mac. Natural associativity and commutativity. Rice Univ. Studie 49 (1963), 28–46.Google Scholar
[9]Leinster, T.. Higher Operads, Higher Categories. London Math. Soc. Lecture Note Series, vol. 298. (Cambridge University Press, 2004). ArXiv:math.CT/0305049.CrossRefGoogle Scholar
[10]Rivano, N. Saavedra. Catégories Tannakiennes. Lecture Notes in Math. 265 (Springer-Verlag, 1972).CrossRefGoogle Scholar
[11]Simpson, C.. Homotopy types of strictc 3-groupoids. Preprint, ArXiv:math.CT/9810059.Google Scholar
[12]Stasheff, J. D.. Homotopy associativity of H-spaces. I, II. Trans. Amer. Math. Soc 108 (1963), 275–292; 293–312.Google Scholar