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The low-dimensional structures formed by tricategories

Published online by Cambridge University Press:  01 May 2009

RICHARD GARNER  and
NICK GURSKI
RICHARD GARNER
Affiliation:
Department of Mathematics, Uppsala University, Box 480, S-751 06 Uppsala, Sweden. e-mail: richard@math.uu.se
NICK GURSKI
Affiliation:
Department of Mathematics, Yale University, 10 Hillhouse Avenue, PO Box 208283, New Haven, CT 06520-8283, U.S.A. e-mail: michaeln.gurski@yale.edu
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Abstract

We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical bicategory, this being a bicategory enriched in the monoidal 2-category of pseudo double categories. Finally, we show that every sufficiently well-behaved locally cubical bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 3 , May 2009 , pp. 551 - 589
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]Bénabou, J. Introduction to bicategories. In Reports of the Midwest Category Seminar, pages 1–77 (Springer, 1967).CrossRefGoogle Scholar
[2]Carboni, A., Kelly, G. M., Walters, R. and Wood, R. Cartesian bicategories II. arXiv preprint 0708.1921.Google Scholar
[3]Carmody, S. Cobordism categories. PhD thesis (Cambridge University, 1995).Google Scholar
[4]Cheng, E. and Gurski, N. The periodic table of n-categories II: degenerate tricategories. arXiv preprint 0706.2307.Google Scholar
[5]Day, B. Biclosed bicategories: localisation of convolution. Unpublished, available as arXiv preprint 0705.3485.Google Scholar
[6]Day, B. and Street, R.Monoidal bicategories and Hopf algebroids. Adv. Math. 129 (1) (1997), 99–157.CrossRefGoogle Scholar
[7]Fiore, T. Pseudo algebras and pseudo double categories. Journal of Homotopy and Related Structures, to appear.Google Scholar
[8]Garner, R.Double clubs. Cahiers Topol. Geom. Différ. Catég. 47 (4) (2006), 261–317.Google Scholar
[9]Gordon, R., Power, A. J. and Street, R.Coherence for tricategories. Mem. Ame. Math. Soc. 117 (558) (1995).Google Scholar
[10]Grandis, M. and Paré, R.Limits in double categories. Cahiers Topol. Geom. Différ. Catég. 40 (3) (1999), 162–220.Google Scholar
[11]Grandis, M. and Paré, R.Adjoints for double categories. Cahiers Topol. Geom. Différ. Catég. 45 (3) (2004), 193–240.Google Scholar
[12]Gurski, N. An algebraic theory of tricategories. PhD thesis (University of Chicago, 2006).Google Scholar
[13]Kelly, G. M. and Lack, S.On property-like structures. Theory Appl. Categ. 3 No. 9 (1997), 213–250 (electronic).Google Scholar
[14]Kelly, G. M. and Street, R. Review of the elements of 2-categories. In Category Seminar (Proc. Sem., Sydney, 1972/1973), pages 75–103. Lecture Notes in Math., vol. 420 (Springer, 1974).CrossRefGoogle Scholar
[15]Lack, S. The algebra of distributive and extensive categories. PhD thesis (Cambridge University, 1995).Google Scholar
[16]Lack, S. Icons. arXiv preprint 0711.4657.Google Scholar
[17]Lack, S. and Paoli, S. 2-nerves for bicategories. K Theory, to appear.Google Scholar
[18]Mac Lane, S. and Paré, R.Coherence for bicategories and indexed categories. J. Pure Appl. Alg. 37 (1) (1985), 59–80.CrossRefGoogle Scholar
[19]Shulman, M. Framed bicategories and monoidal fibrations. arXiv preprint 0706.1286.Google Scholar
[20]Street, R.Enriched categories and cohomology. Repr. Theory Appl. Categ. (14) (2005), 1–18 (electronic). Reprinted from Quaestiones Math. 6 (1983), no. 1-3, 265–283, with new commentary by the author.Google Scholar
[21]Street, R.Weak omega-categories. “Diagrammatic Morphisms and Applications”, Contemp. Math. 318 (2003), 207–213.CrossRefGoogle Scholar
[22]Verity, D. Enriched categories, internal categories, and change of base. PhD thesis (Cambridge University, 1992).Google Scholar
[23]Wood, R. J.Abstract proarrows. I. Cahiers Topol. Geom. Différe. Catég. 23 (3) (1982), 279–290.Google Scholar
[24]Wood, R. J.Proarrows. II. Cahiers Topol. Geom. Différ. Catég. 26 (2) (1985), 135–168.Google Scholar