Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-25T05:51:46.660Z Has data issue: false hasContentIssue false
  • English
  • Français

Homotopy limits for 2-categories

Published online by Cambridge University Press:  01 July 2008

NICOLA GAMBINO
NICOLA GAMBINO*
Affiliation:
Département de Mathématiques, Université du Québec à Montréal, Case Postale 8888, Succursale Centre-Ville, Montréal (Québec) H3C 3P8, Canada. e-mail: nicola.gambino@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using these results, we describe the homotopical behaviour not only of conical limits but also of weighted limits. Finally, pseudo-limits are related to homotopy limits.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 1 , July 2008 , pp. 43 - 63
Copyright
Copyright © Cambridge Philosophical Society 2008

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]Anderson, D. W.. Fibrations and geometric realizations. Bull. Amer. Math. Soc. 84 (5), (1978), 765–788.CrossRefGoogle Scholar
[2]Artin, M., Grothendieck, A. and Verdier, J. L.Théorie des topos et cohomologie etale des schémas (SGA 4), Lecture Notes in Math. vol. 270 (Springer, 1972).Google Scholar
[3]Bird, G. J., Kelly, G. M., Power, A. J. and Street, R. H.. Flexible limits for 2-categories. J. Pure Appl. Alg. 61 (1989), 1–27.CrossRefGoogle Scholar
[4]Blackwell, R., Kelly, G. M. and Power, A. J.. Two-dimensional monad theory. J. Pure Appl. Alg. 59 (1989), 1–41.Google Scholar
[5]Bousfield, A. K. and Kan, D. M.. Homotopy limits, completions and localizations. Lecture Notes in Math vol. 304 (Springer, 1972).Google Scholar
[6]Chachólski, W. and Scherer, J.. Homotopy theory of diagrams. Mem. Amer. Math. Soc. 155 (736) (2002).Google Scholar
[7]Dwyer, W. G., Hirschhorn, P. S., Kan, D. M. and Smith, J. H.Homotopy limit functors on model categories and homotopical categories. Math Surveys Monogr. vol. 113 (2004).Google Scholar
[8]Garner, R.. Cofibrantly generated natural weak factorisation systems. ArXiv:math/0702290, 2007.Google Scholar
[9]Gaunce Lewis, L. Jr., and Mandell, M. A.. Modules in monoidal model categories. J. Pure Appl. Alg. 210 (2007), 395–421.CrossRefGoogle Scholar
[10]Giraud, J.. Cohomologie Non Abélienne. (Springer, 1971).Google Scholar
[11]Gray, J.. Closed categories, lax limits, and homotopy limits. J. Pure Appl. Alg. 19 (1980), 127–158.Google Scholar
[12]Grothendieck, A.. Revêtements étales et groupe fondamental (SGA 1). Lecture Notes in Math. vol. 224 (Springer, 1971).CrossRefGoogle Scholar
[13]Hirschhorn, P.. Model categories and their localizations. Math. Surveys Monogr. vol. 99 (2002).Google Scholar
[14]Hovey, M.. Model categories. Math Surveys Monogr. vol. 63 1998.Google Scholar
[15]Joyal, A. and Tierney, M.. Strong stacks and classifying spaces. In Category Theory (Como 1990), Lecture Notes in Math. vol. 1488 (Springer, 1991), 213–236.Google Scholar
[16]Kelly, G. M.. Doctrinal adjunction. In Kelly and Street [19], pages 257–280.Google Scholar
[17]Kelly, G. M.. Basic concepts of enriched category theory. London Math. Soc. Lecture Note. 64 (Cambridge University Press, 1982. Available online in the Reprints in Theory and Applications of Categories.Google Scholar
[18]Kelly, G. M.. Elementary observations on 2-categorical limits. Bull. Austral. Math. Soc. 39 (2) (1989), 301–317.Google Scholar
[19]Kelly, G. M. and Street, R. H., editors. Category Seminar (Proc. Sem. Sydney 1972/1973), Lecture Notes in Math. vol. 420 (Springer, 1974).Google Scholar
[20]Kelly, G. M. and Street, R. H.. Review of the elements of 2-categories. In Kelly and Street [19], pages 75–103.Google Scholar
[21]Lack, S.. Homotopy-theoretic aspects of 2-monads. ArXiv:math/0607646, 2006. To appear in J. Homotopy and Related Structures.Google Scholar
[22]May, J. P.. The geometry of iterated loop spaces, Lecture Notes in Maths. vol. 271 (Springer, 1972).CrossRefGoogle Scholar
[23]Power, J. and Robinson, E.. A characterisation of pie limits. Math. Proc. Camb. Phil. Soc. 110 (1) (1991), 33–47.CrossRefGoogle Scholar
[24]Rezk, C.. A model category for categories. Available from the author's web page, 1996.Google Scholar
[25]Schwede, S. and Shipley, B.. Algebras and modules in monoidal model categories. Proc. London Math. Soc. 80 (2) (2000), 491–511.CrossRefGoogle Scholar
[26]Shulman, M.. Homotopy limits and colimits and enriched homotopy theory. ArXiv:math/0610194, 2006.Google Scholar
[27]Street, R.. Limits indexed by category-valued 2-functors. J. Pure Appl. Alg. 8 (1976), 149–181.Google Scholar
[28]Thomason, R. W.. Cat as a closed model category. Cah. de Topol. Géom. Différ. XXI (3) (1980), 305–324.Google Scholar