Abelian and Triangulated Categories

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 240)


The purpose of these notes is to provide enough background information to define triangulated categories. We provide background information on monomorphisms, epimorphisms, kernels and cokernels. We use this information to define additive and abelian categories, and provide several examples. We give a full definition of triangulated categories and the axioms TR1TR4. We conclude by describing some basic results on triangulated categories, leading to a long exact sequence of morphism groups.


Abelian Category Cokernel Provide Background Information Morphism Group Monomorphism 
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  1. 1.
    Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1969)Google Scholar
  2. 2.
    Dold, A., Puppe, D.: Homologie nicht-additiver Funktoren. Anwendungen. Ann. Inst. Fourier Grenoble 11, 201–312 (1961).
  3. 3.
    Evans, J.: Tikz code for the octahedral axiom. Accessed: 2016, Published: 2013
  4. 4.
    Freyd, P.: Abelian categories. An introduction to the theory of functors. Harper’s Series in Modern Mathematics. Harper & Row, Publishers, New York (1964)Google Scholar
  5. 5.
    Hartshorne, R.: Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20. Springer-Verlag, Berlin-New York (1966)Google Scholar
  6. 6.
    Holm, T., Jørgensen, P.: Triangulated categories: definitions, properties, and examples. In: Triangulated categories, London Math. Soc. Lecture Note Ser., vol. 375, pp. 1–51. Cambridge Univ. Press, Cambridge (2010).
  7. 7.
    Huybrechts, D.: Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford (2006).
  8. 8.
    Krishnan, V.S.: An introduction to category theory. North-Holland Publishing Co., New York-Amsterdam (1981)Google Scholar
  9. 9.
    Neeman, A.: Triangulated categories, Annals of Mathematics Studies, vol. 148. Princeton University Press, Princeton, NJ (2001).
  10. 10.
    Verdier, J.L.: Des catégories dérivées des catégories abéliennes. Astérisque (239), xii+253 pp. (1997) (1996). With a preface by Luc Illusie, Edited and with a note by Georges MaltsiniotisGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of AlbertaEdmontonCanada

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