Hostname: page-component-54dcc4c588-dbm8p Total loading time: 0 Render date: 2025-09-11T21:03:07.088Z Has data issue: false hasContentIssue false
  • English
  • Français

A homotopy-theoretic universal property of Leinster's operad for weak ω-categories

Published online by Cambridge University Press:  22 May 2009

RICHARD GARNER
RICHARD GARNER*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB. e-mail: rhgg2@cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak ω-categories, showing that the universal and canonical cofibrant replacement of the operad for strict ω-categories is precisely Leinster's operad for weak ω-categories.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 3 , November 2009 , pp. 615 - 628
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.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Batanin, M. A.Monoidal globular categories as a natural environment for the theory of weak n-categories. Adv. in Math. 136 (1998), no. 1, 39–103.CrossRefGoogle Scholar
[2]Batanin, M. A. and Weber, M. Algebras of higher operads as enriched categories. Preprint (2008), available at http://arxiv.org/abs/0803.3594.Google Scholar
[3]Beck, J. Distributive laws. Seminar on Triples and Categorical Homology Theory (ETH, Zürich, 1966/67). Lecture Notes in Math., vol. 80 (Springer, 1969), pp. 119–140.CrossRefGoogle Scholar
[4]Bousfield, A. K.Constructions of factorization systems in categories. J. Pure Appl. Alg. 9 (1977), no. 2-3, 207–220.CrossRefGoogle Scholar
[5]Cheng, E. Monad interleaving: a construction of the operad for Leinster's weak ω-categories. J. Pure Appl. Alg. (2008), in press.Google Scholar
[6]Crans, S.Quillen closed model structures for sheaves. J. Pure Appl. Alg. 101 (1995), 35–57.CrossRefGoogle Scholar
[7]Garner, R. Understanding the small object argument. Appl. Categ. Struct. (2009), in press.Google Scholar
[8]Grandis, M. and Tholen, W.Natural weak factorization systems. Arch. Math. 42 (2006), no. 4, 397–408.Google Scholar
[9]Hirschhorn, P. S. Model categories and their localizations. Math. Sur. Monogr., vol. 99 (American Mathematical Society, 2003).Google Scholar
[10]Hovey, M. Model categories. Math. Surv. Monogr., vol. 63 (American Mathematical Society, 1999).Google Scholar
[11]Johnstone, P. T.Adjoint lifting theorems for categories of algebras. Bull. London Math. Soc. 7 (1975), no. 3, 294–297.CrossRefGoogle Scholar
[12]Kelly, G. M.A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves and so on. Bull. Aust. Math. Soc. 22 (1980), no. 1, 1–83.CrossRefGoogle Scholar
[13]Leinster, T.Operads in higher-dimensional category theory. Theory Appl. Categ. 12 (2004), no. 3, 73–194.Google Scholar
[14]Quillen, D. G. Homotopical algebra. Lecture Notes in Math., vol. 43 (Springer-Verlag, 1967).Google Scholar
[15]Radulescu-Banu, A. Cofibrance and completion. Ph.D. thesis, MIT (1999).Google Scholar
[16]Reedy, Charles, Homotopy theory of model categories. Unpublished note, available at http://www-math.mit.edu/~psh/ (1974).Google Scholar
[17]Rosický, J. and Tholen, W.Lax factorization algebras. J. Pure Appl. Alg. 175 (2002), 355–382.CrossRefGoogle Scholar