nLab element in an abelian category

Idea

Not every abelian category is a concrete category, such as Ab or $R$Mod, hence its objects do not necessarily 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.

This is closely related to embedding the abelian category fully-faithfully and exactly into a concrete abelian category. See at abelian category the section Embedding theorems for more on this.

Definition

One definition goes likes this (MacLane, VIII.4 and also for instance Gelfand-Manin, II.5):

An element of an object $W$ in a given abelian category $\mathcal{A}$ is an equivalence class $[X,h]$ of pairs $(X,h)$ where $X$ is an object of $\mathcal{A}$ and $h:X\to W$ a morphism (hence a generalized element) and the equivalence is defined as follows: $[X,h] = [Y,k]$ iff there exists an object $Z$ in $\mathcal{A}$ and epimorphisms $u:Z\to X$, $v:Z\to Y$ such that $h\circ u = k\circ v : Z\to W$.

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 1974 where the actual generalized elements are remembered but a refinement of their domain is allowed, much as familiar from topos theory.

References

Equivalence classes of generalized elements are considered for instance in

• Saunders Mac Lane, Categories for the working mathematician, 2nd edition, Springer 1998.

• Sergei Gelfand, Yuri Manin, Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original. Springer 1996. xviii+372 pp.; 2nd corrected ed. 2002.

Genuine generalized elements are considered in

• George Bergman, A note on abelian categories – translating element-chasing proofs, and exact embedding in abelian groups (1974) [pdf, pdf]