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Homotopy theory of well-generated algebraic triangulated categories

Published online by Cambridge University Press:  11 February 2008

Gonçalo Tabuada
Gonçalo Tabuada
Affiliation:
Université Paris 7 – Denis Diderot, UMR 7586 du CNRS, case 7012, 2 Place Jussieu, 75251 Paris cedex 05, France, tabuada@math.jussieu.fr.
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Abstract

For every regular cardinal α, we construct a cofibrantly generated Quillen model structure on a category whose objects are essentially dg categories which are stable under suspensions, cosuspensions, cones and α-small sums.

Using results of Porta, we show that the category of well-generated (algebraic) triangulated categories in the sense of Neeman is naturally enhanced by our Quillen model category.

Keywords

Monads and algebrasQuillen model structurewell-generated algebraic triangulated categoryVerdier's quotientexact dg categoryα-cocompletness
Type
Research Article
Information
Journal of K-Theory , Volume 3 , Issue 1 , February 2009 , pp. 53 - 75
Copyright
Copyright © ISOPP 2009

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References

1.Carboni, A. and Johnstone, P.Connected limits, familial representability and Artin glueing, Fifth Biennial Meeting on Category Theory and Computer Science (Amsterdam, 1993), Math. Structures Comput. Sci., volume 5, 1995, 4, 441–459CrossRefGoogle Scholar
2.Borceux, F., Handbook of categorical algebra. 2, Encyclopedia of Mathematics and its Applications, 51, Cambridge University Press, 1994Google Scholar
3.Ciesielski, K., Set theory for the working mathematician, London Mathematical Society Student Texts, 39, Cambridge University Press, 1997Google Scholar
4.Drinfeld, V., DG quotients of DG categories, Journal of Algebra, 272, 2004, 643–691CrossRefGoogle Scholar
5.Goerss, P. and Jardine, J., Simplicial homotopy theory, Progress in Mathematics, 174, 1999Google Scholar
6.Hirschhorn, P., Model categories and their localizations, Mathematical Surveys and Monographs, 99, American Mathematical Society, 2003Google Scholar
7.Keller, B., On differential graded categories, International Congress of Mathematicians Madrid 2006, Vol II, 151–190Google Scholar
8.Keller, B., On differential graded categories, I.C.M. talk 2006, MadridGoogle Scholar
9.Keller, B. (joint with M. Porta), Well generated triangulated categories, Oberwalfch report, 8, 2006Google Scholar
10.Kelly, G. and Lack, S., On property-like structures, Theory and Applications of Categories, 3, 1997, 213–250Google Scholar
11.Kontsevich, M., Triangulated categories and geometry, Course at E.N.S. Paris, 1998Google Scholar
12.Kontsevich, M., Topological field theory for triangulated categories, Talk at the conference on K-theory and Noncommutative Geometry, Institut Henri Poincaré, Paris, 06 2004Google Scholar
13.Krause, H., On Neeman's well generated triangulated categories, Documenta Mathematica, 6, 2001, 121–126CrossRefGoogle Scholar
14.Maclane, S., Categories for the working mathematician, Graduate Texts in Mathematics, 5, Springer-Verlag, 1998Google Scholar
15.Neeman, A., Triangulated categories, Annals of Mathematics Studies, 148, Princeton University Press, 2001Google Scholar
16.Neeman, A., On the derived category of sheaves on a manifold, Documenta Mathematica, 6, 2001, 483–488CrossRefGoogle Scholar
17.Popesco, N. and Gabriel, P., Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes, C. R. Acad. Sci. Paris 258, (1964), 4188–4190Google Scholar
18.Porta, M., Ph.D. thesis in preparationGoogle Scholar
19.Quillen, D., Homotopical algebra, Lecture Notes in Mathematics, 43, Springer-Verlag, 1967Google Scholar
20.Tabuada, G., A new Quillen model for the Morita homotopy theory of DG categories, preprint math.KT/0701205Google Scholar
21.Tabuada, G., Addendum à Invariants additifs de dg-catégories, Int. Math. Res. Notices 26 (2006)Google Scholar
22.Tabuada, G., Invariants additifs de dg-catégories, Int. Math. Res. Notices 53(2005), 3309–3339CrossRefGoogle Scholar
23.Tabuada, G., Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Math. Acad. Sci. Paris, 340, 2005, 15–19CrossRefGoogle Scholar
24.Toën, B., The homotopy theory of dg-categories and derived Morita theory, Inventiones Mathematicae, 167, (2007), 615–667CrossRefGoogle Scholar
25.Verdier, J-L., Des catégories dérivées des catégories abeliennes, Astérisque, 239, 1996Google Scholar