Homological Algebra/Abelian Category

Definition ($Ab$-enriched category):

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

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

Definition (zero object):

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

Definition (biproduct):

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

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

We usually denote $c$ by $a\oplus b$.

Definition (additive category):

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

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

Definition ((co-)kernel):

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

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 $R$-modules of a ring $R$ is an abelian category.

Exercises

1. Given $f:a\to b$ in an $Ab$-enriched category with zero object. Prove that $f=0_{a,b}$ iff $f$ factors through $\mathbf {0}$.

1. Given a biproduct $(a\oplus b,i_{1},p_{1},i_{2},p_{2})$ of $a$ and $b$. Prove that $(a\oplus b,i_{1},i_{2})$ is a coproduct of $a$ and $b$ and $(a\oplus b,p_{1},p_{2})$ is a product of $a$ and $b$.

1. In an $Ab$-enriched category with zero object, a kernel of $f:a\to b$ can be equivalently be characterized as a pullback of $\mathbf {0} \to b$ along $f$.