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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

Direct Limit Abelian Category Left Adjoint Canonical Morphism Exact Functor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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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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