Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T17:22:08.112Z Has data issue: false hasContentIssue false
  • English
  • Français

EQUIVALENCE OF MODELS FOR EQUIVARIANT (∞, 1)-CATEGORIES

Part of: Homotopy theory Applied homological algebra and category theory Homological algebra

Published online by Cambridge University Press:  10 June 2016

JULIA E. BERGNER
JULIA E. BERGNER*
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521, USA e-mail: julia.bergner@ucr.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we show that the known models for (∞, 1)-categories can all be extended to equivariant versions for any discrete group G. We show that in two of the models we can also consider actions of any simplicial group G.

MSC classification

Primary: 55U35: Abstract and axiomatic homotopy theory
Secondary: 55P91: Equivariant homotopy theory 55U40: Topological categories, foundations of homotopy theory 18G55: Homotopical algebra
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 1 , January 2017 , pp. 237 - 253
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Bergner, J. E., A characterization of fibrant Segal categories, Proc. Amer. Math. Soc. 135 (2007), 4031–4037.Google Scholar
2. Bergner, J. E., A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007), 2043–2058.Google Scholar
3. Bergner, J. E., Three models for the homotopy theory of homotopy theories, Topology 46 (2007), 397–436.Google Scholar
4. Bergner, J. E. and Chadwick, S. G., Equivariant complete Segal spaces, Homology, Homotopy Appl. 17 (2), (2015), 371–381.Google Scholar
5. Bohmann, A. M., Mazur, K., Osorno, A., Ozornova, V., Ponto, K. and Yarnall, C., A model structure on $G \mathcal Cat$ , in Women in topology: collaborations in homotopy theory, Contemp. Math., vol. 641 (American Mathematical Society, Providence, RI, 2015), 123–134.Google Scholar
6. Cisinski, D.-C. and Moerdijk, I., Dendroidal Segal spaces and ∞-operads, J. Topol. 6 (3) (2013), 675–704.CrossRefGoogle Scholar
7. Cisinski, D.-C. and Moerdijk, I., Dendroidal sets and simplicial operads, J. Topol. 6 (3) (2013), 705–756.Google Scholar
8. Cisinksi, D.-C. and Moerdijk, I., Dendroidal sets as models for homotopy operads, J. Topol. 4 (2) (2011), 257–299.Google Scholar
9. Cordier, J. M. and Porter, T., Vogt's theorem on categories of homotopy coherent diagrams, Math. Proc. Camb. Phil. Soc. 100 (1986), 65–90.Google Scholar
10. Dotto, E. and Moi, K., Homotopy theory of G-diagrams and equivariant excision, Algebr. Geom. Topol. 16 (1) (2016), 325–395.Google Scholar
11. Dugger, D. and Spivak, D. I., Rigidification of quasicategories, Algebr. Geom. Topol. 11 (2011), 225–261.Google Scholar
12. Dyckerhoff, T. and Kapranov, M., Higher Segal spaces I, preprint available at math.AT/1212.3563.Google Scholar
13. Elmendorf, A. D., Systems of fixed point sets. Trans. Amer. Math. Soc. 277 (1) (1983), 275–284.Google Scholar
14. Hirschhorn, P. S., Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99 (AMS, 2003).Google Scholar
15. Joyal, A., Simplicial categories vs quasi-categories, in preparation.Google Scholar
16. Joyal, A., The theory of quasi-categories I, in preparation.Google Scholar
17. Joyal, A. and Tierney, M., Quasi-categories vs Segal spaces, Contemp. Math. 431 (2007), 277–326.Google Scholar
18. Lack, S., Re: Classifying functor and colimits, Category theory list, http://permalink.gmane.org/gmane.science.mathematics.categories/3407.Google Scholar
19. Lurie, J., Higher topos theory, Annals of Mathematics Studies, vol. 170 (Princeton University Press, Princeton, NJ, 2009).Google Scholar
20. Lurie, J., On the classification of topological field theories, in Current developments in mathematics, 2008 (Int. Press, Somerville, MA, 2009), 129–280.Google Scholar
21. Pelissier, R., Catégories enrichies faibles, preprint available at math.AT/0308246.Google Scholar
22. Piacenza, R. J., Homotopy theory of diagrams and CW-complexes over a category, Canad. J. Math. 43 (4) (1991), 814–824.CrossRefGoogle Scholar
23. Reedy, C. L., Homotopy theory of model categories, unpublished manuscript, available at http://www-math.mit.edu/~psh.Google Scholar
24. Rezk, C., A cartesian presentation of weak n-categories, Geom. Topol. 14 (2010), 521–571.Google Scholar
25. Rezk, C., A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (3), 973–1007.Google Scholar
26. Stephan, M., On equivariant homotopy theory for model categories, Homology, Homotopy Appl. preprint available at math.AT/1308.0856.Google Scholar