Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Left-determined model categories and universal homotopy theories

Authors: J. Rosicky and W. Tholen
Journal: Trans. Amer. Math. Soc. 355 (2003), 3611-3623
MSC (2000): Primary 55U35
Published electronically: May 15, 2003
Erratum: Trans. Amer. Math. Soc. 360 (2008), 6179-6180
MathSciNet review: 1990164
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We say that a model category is left-determined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not left-determined, we show that its non-oriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural left-determined model category structure. This is used to give another and, as we believe simpler, proof of a recent result of D. Dugger about universal homotopy theories.

References [Enhancements On Off] (What's this?)

  • [AHRT1] J. Adámek, H. Herrlich, J. Rosický, and W. Tholen, On a generalized small-object argument for the injective subcategory problem, Cahiers Topologie Géom. Différentielle Catég. XLIII (2002), 83-106.
  • [AHRT2] J. Adámek, H. Herrlich, J. Rosický, and W. Tholen, Weak factorization systems and topological functors, Applied Categ. Structures 10 (2002), 237-249.
  • [AR] Jiří Adámek and Jiří Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series, vol. 189, Cambridge University Press, Cambridge, 1994. MR 1294136
  • [B] Tibor Beke, Sheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 447–475. MR 1780498,
  • [CSS] C. Casacuberta, D. Sceveneles, and J. H. Smith, Implications of large-cardinal principles in homotopical localizations, preprint, 1998.
  • [D] Daniel Dugger, Universal homotopy theories, Adv. Math. 164 (2001), no. 1, 144–176. MR 1870515,
  • [G] Marco Grandis, Higher fundamental functors for simplicial sets, Cahiers Topologie Géom. Différentielle Catég. 42 (2001), no. 2, 101–136 (English, with French summary). MR 1839359
  • [H] P. S. Hirschhorn, Localization of Model Categories, preprint, 1998,
  • [Ho] Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
  • [EZ] J. Eilenberg and J. A. Zilber, Semi-simplicial complexes and singular homology, Ann. of Math. 51 (1950), 499-513. MR 11:734e
  • [L] F. W. Lawvere, Toposes generated by codiscrete objects, in Combinatorial Topology and Functional Analysis, Notes 1988, 1989, 1992.
  • [Q] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
  • [RT] J. Rosický and W. Tholen, Lax factorization algebras, J. Pure Appl. Algebra 175 (2002), 355-382.
  • [S] J. H. Smith, Combinatorial model categories, in preparation.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 55U35

Retrieve articles in all journals with MSC (2000): 55U35

Additional Information

J. Rosicky
Affiliation: Department of Mathematics, Masaryk University, 662 95 Brno, Czech Republic

W. Tholen
Affiliation: Department of Mathematics and Statistics, York University, Toronto M3J 1P3, Canada

Received by editor(s): June 1, 2002
Published electronically: May 15, 2003
Additional Notes: The first author was supported by the Grant Agency of the Czech Republic under Grant 201/99/0310. The hospitality of the York University is gratefully acknowledged.
The second author was supported by the Natural Sciences and Engineering Council of Canada
Article copyright: © Copyright 2003 American Mathematical Society