Grothendieck Categories

Abstract

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

• AB3. A coproductive, hence cocomplete, abelian category.

• 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.

• AB5. An AB3 category such that for any directed family {A _i } _{i ∈I} of subobjects of any object X, and any subobject B,

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