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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: [email protected]
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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.

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

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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