I'm trying to learn how to perform diagram-chasing in abstract Abelian categories. Instead of an approach with some elements one have to use universal properties somehow in the proof. But I reckon the need of some lemmas...
A category is Abelian if:
- it has a zero object
- it has all binary products and binary coproducts
- it has all kernels and cokernels
- every monomorphism is a kernel to some morphism
- every epimorphism is a cokernel to some morphism
The first "lemma" that coming into my mind (though I don't really know if it is true) deals with the connection between being monic and have a kernel equal to zero.
From the universal properties:
(1)$\quad f$ is a monomorphism if it given morphisms $m,n$ holds that $\beta m=\beta n\implies m=n$ $\require{AMScd}$ \begin{CD} X @>m>n> B@>\beta>> B' \end{CD} (2)$\quad k$ is a kernel to $\beta:B\to B'$ if $\beta k=0$ and for each $k'$ with $\beta k'=0$ there is a unique morphism $\phi$ such that $k\phi=k'$ \begin{CD} K'@>k'>>B\\ @V\exists!\phi VV\# @|\\ K @>k>> B@>\beta>> B' \end{CD}
How to prove that $\operatorname{ker}\beta=0\implies \beta$ is mono, using (1) and (2)?
My own approach, unfortunately, consists of staring on the diagrams above.