How to prove that the category of Abelian groups is abelian category?
I use the following definition of abelian category.
(1) The category is $\mathsf{Ab}$-enriched category.
(2) The category has all finite limits and colimits.
(3) Every monic is a kernel of some morphism and every epic is a cokernel of some morphism.
It is easy to check (1), but (2), (3) is hard for me. How do I prove?
I really need an explicit proof of (2) because I need some concrete foundations to understand category theory. When the things are required to have all finite limits and colimits, how to deal with limits and colimits in real proof.
I guess that (3) is an result of commutative algebra.
p.s. Maybe, using other equivalent definitions is more easy to prove it. However, I couldn't use them yet because I haven't proven that these definitions are equivalent.
p.s. 2 I tried to find some references about it, but 'Abelian'(duplicates to Category of 'Abelian' groups and 'Abelian' Category) makes me hard to search.
For (2), follow the steps (note that the ideas are apply more generally than just the concrete context of Ab):
a. A category which has finite products and equalizers has finite limits. The idea is to write the limit of any diagram in terms of products and equalizers. By Duality Principle, you have the corresponding result for finite colimits.
b. If a category is Ab-enriched, then finite products are the same as finite coproducts. Thus, Ab in particular, has finite (co)products (a zero object, and binary (co)products).
c. Equalizers and coequalizers in an Ab-category are the same as kernels and cokernels. More precisely, $Eq(f,g) = Kernel(f-g)$ and $Kernel(f) = Eq(f,0)$. Same for the dual notions. Now, Ab certainly has kernels. For the cokernel of a morphism $f : A \rightarrow B$ in Ab, consider the canonical map $B \rightarrow B/Im f$.
Steps a-c show that $Ab$ is finitely complete and cocomplete. This proves criterion (2).
For (3), check that
a. A monomorphism in the category Ab is the same as an injection. An epimorphism in Ab is the same as a surjection.
b. If $f : A \rightarrow B$ is a monomorphism in Ab, then $f$ is the kernel of its cokernel $B \rightarrow B/f(A)$. Also, if $f$ is an epimorphism in Ab, then $f$ is the cokernel of its kernel $ker(f) \hookrightarrow A$ (you will be using the First Isomorphism Theorem here).
This proves (3). Note that (b) is actually true in any abelian category. Once you assume that monics are normal(kernels of maps) and epics are conormal, you can actually show in an abelian category that every monic is the kernel of its cokernel and dually, every epic is the cokernel of its kernel.