Homological Algebra/Defintion of abelian category

Definition (${\displaystyle Ab}$-enriched category):

An ${\displaystyle Ab}$-enriched category is a category ${\displaystyle {\mathcal {C}}}$ such that:

1. ${\displaystyle \forall a,b\in \operatorname {Obj} ({\mathcal {C}})}$, ${\displaystyle {\mathcal {C}}(a,b)}$ is an abelian group.
2. ${\displaystyle \forall a,b,c\in \operatorname {Obj} ({\mathcal {C}})}$, ${\displaystyle \circ _{a,b,c}:{\mathcal {C}}(b,c)\times {\mathcal {C}}(a,b)\to {\mathcal {C}}(a,c)}$ is bilinear.

Definition (zero object):

A zero object is an object in an ${\displaystyle Ab}$-enriched category that is both initial and terminal. We usually denote it by ${\displaystyle \mathbf {0} }$.

Definition (biproduct):

Given an ${\displaystyle Ab}$-enriched category ${\displaystyle {\mathcal {C}}}$, a biproduct of ${\displaystyle a,b\in \operatorname {Obj} ({\mathcal {C}})}$ is a tuple ${\displaystyle (c,i_{1}:a\to c,p_{1}:c\to a,i_{2}:b\to c,p_{2}:c\to b)}$ such that:

1. ${\displaystyle p_{1}\circ i_{1}=1_{a}.}$
2. ${\displaystyle p_{2}\circ i_{1}=0_{a,b}.}$
3. ${\displaystyle p_{1}\circ i_{2}=0_{b,a}.}$
4. ${\displaystyle p_{2}\circ i_{2}=1_{b}.}$
5. ${\displaystyle i_{1}\circ p_{1}+i_{2}\circ p_{2}=1_{c}.}$

We usually denote ${\displaystyle c}$ by ${\displaystyle a\oplus b}$.

An additive category is an ${\displaystyle Ab}$-enriched category ${\displaystyle {\mathcal {C}}}$ such that:

1. There is a zero product in ${\displaystyle {\mathcal {C}}}$.
2. Every ${\displaystyle a,b\in \operatorname {Obj} ({\mathcal {C}})}$ has a biproduct.

Definition ((co-)kernel):

Given ${\displaystyle f:a\to b}$ in an ${\displaystyle Ab}$-enriched category. A (co-)kernel of ${\displaystyle f}$ is a (co-)equalizer of ${\displaystyle f}$ and ${\displaystyle 0_{a,b}}$.

Definition (abelian category):

An abelian category is an additive category where:

1. Every morphism has a kernel and cokernel.
2. Every monomorphism is a kernel and every epimorphism is a cokernel.

Example:

The category of all left ${\displaystyle R}$-modules of a ring ${\displaystyle R}$ is an abelian category.

Exercises

1. Given ${\displaystyle f:a\to b}$  in an ${\displaystyle Ab}$ -enriched category with zero object. Prove that ${\displaystyle f=0_{a,b}}$  iff ${\displaystyle f}$  factors through ${\displaystyle \mathbf {0} }$ .
1. Given a biproduct ${\displaystyle (a\oplus b,i_{1},p_{1},i_{2},p_{2})}$  of ${\displaystyle a}$  and ${\displaystyle b}$ . Prove that ${\displaystyle (a\oplus b,i_{1},i_{2})}$  is a coproduct of ${\displaystyle a}$  and ${\displaystyle b}$  and ${\displaystyle (a\oplus b,p_{1},p_{2})}$  is a product of ${\displaystyle a}$  and ${\displaystyle b}$ .
1. In an ${\displaystyle Ab}$ -enriched category with zero object, a kernel of ${\displaystyle f:a\to b}$  can be equivalently be characterized as a pullback of ${\displaystyle \mathbf {0} \to b}$  along ${\displaystyle f}$ .