Algebra pp 486-497 | Cite as

Grothendieck Categories

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


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,
    $$\left( {\sum\limits_{i \in I} {{A_i}} } \right) \cap B = \sum\limits_{i \in I} {\left( {{A_i} \cap B} \right)}.$$


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Copyright information

© Springer-Verlag, Berlin · Heidelberg 1973

Authors and Affiliations

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

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