Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-10T12:30:53.032Z Has data issue: false hasContentIssue false
  • English
  • Français

MODEL STRUCTURES ON TRIANGULATED CATEGORIES

Part of: Homological algebra Abelian categories Applied homological algebra and category theory

Published online by Cambridge University Press:  18 December 2014

XIAOYAN YANG
XIAOYAN YANG*
Affiliation:
Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China e-mail: yangxy@nwnu.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We define model structures on a triangulated category with respect to some proper classes of triangles and give a general study of triangulated model structures. We look at the relationship between these model structures and cotorsion pairs with respect to a proper class of triangles on the triangulated category. In particular, we get Hovey's one-to-one correspondence between triangulated model structures and complete cotorsion pairs with respect to a proper class of triangles. Some applications are given.

MSC classification

Primary: 18E30: Derived categories, triangulated categories
Secondary: 55U35: Abstract and axiomatic homotopy theory 18G25: Relative homological algebra, projective classes

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 2 , May 2015 , pp. 263 - 284
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Asadollahi, J. and Salarian, Sh., Gorenstein objects in triangulated categories, J. Algebra 281 (2004), 264–286.CrossRefGoogle Scholar
2.Beilinson, A. A., Bernstein, J. and Deligne, P., Perverse sheaves, in Analysis and topology on singular spaces I. (Luminy, 1981); Astérisque 100 (1982), 5–171.Google Scholar
3.Beligiannis, A., Relative homological algebra and purity in triangulated categories J. Algebra 227 (2000), 268–361.CrossRefGoogle Scholar
4.Bühler, T., Exact categories, Expo. Math. 28 (2010), 1–69.Google Scholar
5.Gillespie, J., Model structures on exact categories J. Pure Appl. Algebra 215 (2011), 2892–2902.CrossRefGoogle Scholar
6.Hartshorne, R., Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck given at Harvard 1963/64, Lecture Notes in Math., vol. 20 (Springer-Verlag, Berlin, Germany, 1966.CrossRefGoogle Scholar
7.Hovey, M., Model categories (American Mathematical Society, Providence, RI, 1999.Google Scholar
8.Hovey, M., Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 (2002), 553–592.Google Scholar
9.Hubery, A., Notes on the octahedral axiom. Available at http://www.maths.leeds.ac.uk/ahubery/octahedral.pdf.Google Scholar
10.Iversen, B., Octahedra and braids Bull. Soc. Math. France 114 (1986), 197–213.Google Scholar
11.Murfet, D., Triangulated categories part I, April 11, 2007. Available at Therisingsea.org/notes/TriangulatedCategories.pdf.Google Scholar
12.Neeman, A., Triangulated categories, Ann. Math. Stud., vol. 148, (Princeton University Press, 2001).CrossRefGoogle Scholar
13.Puppe, D., On the structure of stable homotopy theory, Colloquium on algebraic topology (Aarhus Universitet Matematisk Institut 1962) 65–71.Google Scholar