Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-30T20:32:12.309Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
NICK GURSKI
Affiliation:
Department of Mathematics, Yale University, 10 Hillhouse Avenue, PO Box 208283, New Haven, CT 06520-8283, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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 Unpublished, available as arXiv preprint 0705.3485.Google Scholar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