/sci/ - Science & Math

>>8960417
Every diagram lemma holds. The proof is quite simple: consider any diagram lemma, and construct an associated diagram in some abelian category [math]\textbf{A}[/math]. The class of objects in the diagram is a set, and this gives rise to an abelian, exact, full and small subcategory [math]\textbf{A}'[/math] of [math]\textbf{A}[/math] such that the set is a subset of this subcategory's set of objects. Now, there is a unitary ring [math]R[/math] such that there is an exact and full embedding [math]M\colon \textbf{A}' \to R \text{-} \textbf{mod}[/math]. Since the diagram lemma holds in the module category, it also holds in [math]\text{A}'[/math]. That it holds in the whole category [math]\textbf{A}[/math] follows from the way the subcategory was constructed.

You should check the metatheorems for abelian categories and Mitchell's embedding theorem to fill in the details.