Abstract
We show that the quasicategory of frames of a cofibration category, introduced by the second-named author, is equivalent to its simplicial localization.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Avigad, J., Kapulkin, K., Lumsdaine, P.L.: Homotopy limits in type theory. Math. Structures Comput. Sci. 25(5), 1040–1070 (2015)
Barwick, C., Kan, D.M.: A characterization of simplicial localization functors and a discussion of DK equivalences. Indag. Math. (N.S.) 23(1-2), 69–79 (2012)
Barwick, C., Kan, D.M.: Relative categories: another model for the homotopy theory of homotopy theories. Indag. Math. (N.S.) 23(1-2), 42–68 (2012)
Cisinski, D.-C.: Catégories dérivables. Bull. Soc. Math. France 138(3), 317–393 (2010)
Cordier, J.-M.: Sur la notion de diagramme homotopiquement cohérent. Cahiers Topologie Géom. Différentielle 23 (1), 93–112 (1982). Third Colloquium on Categories, Part VI (Amiens 1980)
Dwyer, W.G., Hirschhorn, P.S., Kan, D.M., Smith, J.H.: Homotopy limit functors on model categories and homotopical categories, vol. 113. American Mathematical Society, Providence, RI (2004)
Dwyer, W.G., Kan, D.M.: Calculating simplicial localizations. J. Pure Appl. Algebra 18(1), 17–35 (1980)
Dwyer, W.G., Kan, D.M.: Function complexes in homotopical algebra. Topology 19(4), 427–440 (1980)
Dwyer, W.G., Kan, D.M.: Simplicial localizations of categories. J. Pure Appl. Algebra 17(3), 267–284 (1980)
Dugger, D., Spivak, D.I.: Mapping spaces in quasi-categories. Algebr. Geom. Topol. 11(1), 263–325 (2011)
Goerss, P.G., Jardine, J.F.: Simplicial homotopy theory, Progress in Mathematics, vol. 174. Birkhäuser Verlag, Basel (1999). doi:10.1007/978-3-0348-8707-6
Gabriel, P., Zisman, M.: Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer, New York (1967)
Joyal, A.: The theory of quasi-categories and its applications, Vol. II of course notes from Simplicial Methods in Higher Categories, Centra de Recerca Matemàtica, Barcelona (2008, 2009). http://www.crm.es/HigherCategories/notes.html
Joyal, A., Tierney, M.: Quasi-categories vs Segal spaces, Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI, pp. 277–326 (2007)
Kapulkin, K.: Locally cartesian closed quasicategories from type theory, preprint. arXiv:1507.02648 (2015)
Latch, D.M., Thomason, R.W., Wilson, W.S.: Simplicial sets from categories. Math. Z 164(3), 195–214 (1979)
Lurie, J.: Higher topos theory. Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009)
Mac Lane, S.: Categories for the working mathematician, 2 ed. Graduate Texts in Mathematics, vol. 5. Springer, New York (1998)
Rădulescu-Banu, A.: Cofibrations in homotopy theory, preprint. arXiv:math/0610009 (2009)
Rezk, C.: A model for the homotopy theory of homotopy theory. Trans. Amer. Math. Soc. 353(3), 973–1007 (2001). (electronic)
Riehl, E., Verity, D.: The theory and practice of Reedy categories. Theory Appl. Categ. 29, 256–301 (2014)
Szumiło, K.: Two models for the homotopy theory of cocomplete homotopy theories, Ph.D. thesis, University of Bonn. arXiv:1411.0303 (2014)
Toën, B.: Vers une axiomatisation de la théorie des catégories supérieures. K-Theory 34 3, 233–263 (2005)
Vogt, R.M.: The HELP-lemma and its converse in Quillen model categories. J. Homotopy Relat. Struct. 6(1), 115–118 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kapulkin, K., Szumiło, K. Quasicategories of Frames of Cofibration Categories. Appl Categor Struct 25, 323–347 (2017). https://doi.org/10.1007/s10485-015-9422-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-015-9422-y