(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
Not every abelian category is a concrete category, such as Ab or Mod, hence its objects do not necessrily have underlying sets whose elements one can reason about. To circumvent this and reason by “diagram chases” of elements in general abelian categories, one can instead use generalized elements in a suitable way.
An element of an object in a given abelian category is an equivalence class of pairs where is an object of and a morphism (hence a generalized element) and the equivalence is defined as follows: iff there exists an object in and epimorphisms , such that .
However, beware that the passage to equivalence classes does not respect the abelian group structure and hence generalized elements in this sense cannot be added or subtracted. A more natural approach is discussed in (Bergman) where the actual generalized elements are remembered but a refinement of their domain is allowed, much as familiar from topos theory.
Equivalence classes of generalized elements are considered for instance in
Genuine generalized elements are considered in