Algebra pp 486-497 | Cite as

Grothendieck Categories

  • Carl Faith
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 190)

Abstract

Grothendieck [57] introduced the notation for abelian categories which follows:
  1. AB3.

    A coproductive, hence cocomplete, abelian category.

     
  2. AB4.

    An AB3 category in which every coproduct i I A i i I B i of monics {A i B i } i I is monic.

     
  3. AB5.
    An AB3 category such that for any directed family {A i } i I of subobjects of any object X, and any subobject B,
    $$\left( {\sum\limits_{i \in I} {{A_i}} } \right) \cap B = \sum\limits_{i \in I} {\left( {{A_i} \cap B} \right)}.$$
     

Keywords

Hull Dition Mora Proal Prool 

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References

  1. [68]
    Bucur, I., Deleanu, A.: Introduction to the Theory of Categories and Functors. London/New York/Sydney: Wiley-Interscience 1968.Google Scholar
  2. [60]
    Freyd, P.: Functor Theory. Ph. D. Thesis, Columbia University 1970.Google Scholar
  3. [64]
    Freyd, P.: Abelian Categories. New York: Harper Row 1964.Google Scholar
  4. [62]
    Gabriel, P.: Des catégories abeliennes. Bull. Soc. Math. France 90, 323–448 (1962).MATHMathSciNetGoogle Scholar
  5. [57]
    Grothendieck, A.: Sur quelques points d’algèbre homologique. Tohoku Math. J. 9, 119–221 (1957)MATHMathSciNetGoogle Scholar
  6. [60]
    Lubkin, S.: Imbedding of abelian categories. Trans. Amer. Math. Soc. 97, 410–417 (1960).CrossRefMathSciNetGoogle Scholar
  7. [63]
    MacLane, S.: Homology. Berlin/Göttingen/Heidelberg: Springer 1963.MATHGoogle Scholar
  8. [72]
    MacLane, S.: Categories for the working mathemacian. Berlin/Heidelberg/ New York: Springer 1972. MacLane, S., Eilenberg, S. (see Eilenberg and MacLane).Google Scholar
  9. [64]
    Popesco, N., Gabriel, P.: Caractérisations des catégories abelienncs avec générateurs et limites inductives exactes. C. R. Acad. Sci. Paris 258, 4188–4190 (1964). Preston, G. B., Clifford, A. H. (see Clifford and Preston).Google Scholar
  10. [65]
    Mitchell, B.: Theory of Categories. New York: Academic Press 1965.MATHGoogle Scholar
  11. [71]
    Takeuchi, M.: A simple proof of Gabriel and Popesco’s theorem. J. Algebra 18 112–113 (1971).CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag, Berlin · Heidelberg 1973

Authors and Affiliations

  • Carl Faith
    • 1
  1. 1.Rutgers UniversityNew BrunswickUSA

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