Pseudo-element

Summary

A pseudo-element of an object $A$ in an abelian category is an equivalence class of morphisms with codomain $A$. Two such morphisms are identified when they agree after composing with suitable epimorphisms. Pseudo-elements behave enough like elements of a module to allow diagram chasing inside any abelian category. With them, one proves diagram lemmas such as the snake lemma without an embedding theorem.